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QOHROMANOV POI^/ID FORRUX o(luA$BAROY KTRAN I$,,qM og,asoFaRLl ILSAR SEYFODDIN ograHuSEYNOVZAFARQaTARofu
(HOLLi ir,e)
DARS VOSAiTi
&a\
Ehni-MetdiW $uras "Rlnzi1t1w" blmasinin(09.06.2009-ca il 3 vyh prototoh) tavsiyyava ruzirin 07.07.2009-ca il 858 wyh amri iladars uasaiti kimi tasdiq edilmi$ir.
It,.fI
SUMQAYTT 2009
RTYAZI ANALI'LDONMOSOLO VO MISALLAR
I IilSSO
5/{ 2A ffraom* veranlar: prof Hi.Aslanov
Prof. F.G.FeYziYev
Elmi rcdohtorlal' dos. N.T.Qurbanov
dos X,H AltYev
Redthtor: dos- V.A.MustafaYev
Dors vasditi Tehsil Nszirliyi tarafindan tasdiq olunmut " RiWionahz" fow oroqramr esasnda brtib olunub va tam htrsu ahata edir'-
Oais vaiain'nyziyyat va fizifu faH)halarttda tahsil al@ telebalzr
gfrn n zarda tutulmuldur.
ciRigRiyazi amJiz fenni universitetin fiaka-riyaziTy at fakilltalerinda tedris
olunan asas fandir. Bu fannin baqlanlrcrnda goxlu yeni anlayqlarverilmasi, baSqa riyazi fanlara nisbatan bu kursun menimsenilmasini birqadar gatinlegdirir. Umumiyysile, riyazi mrliz kursunun yax$l
manimssnilmesine mane olan sebablsrdan biri de onun baglanf,rcmmsoruakr hissalare nisbeten daha mtlccrrad olmasrdrr. Bu gatinlikteri aradanqaldrmaq og{ln talabalsrin praktik tacr[basi olmahdr. Bu msqssdlateqdim etdiyimiz dars vesaitinde bir 9ox mdvzulara aid misallann helliizahh verilmiqdir.
Qalrgm4rq ki, riyazi analiz kursuna aid mesala ve misallann hellitasdiq olunmug mdvcud proqramr ehab etsin ve universitetlora yeni daxilolan talebelar terafmden sarbast oxunsun.
Aydrndr ki bu saheni ahate edan Azarbaycan dilinds derslik ve darsvesaitinin kifayet qsder az olmasr bu ders vasaitinin telabelerin istifadasineverilmasi mOvcud alabatdan irali galir.
Dars vasaitinda milash telablr nezara alroaraq I fasilde bozibarabarsizliklerin isbatr yeni iisullarla verilmiydir. Funksiyalarmlimitlarinin hesablanrnasrnda istifade olunan sonlu sayda adsdlerinistonilan tsrtibdan camlerinin taprlmasr ugiln dustrlarm alrnmasr yeniilsullarla veriJrnigdir.
tr fesilds diferensial hesabmtn biltih teoremleri U90n tapgrrq hstli ilaverilmig va funksiyan m rodqiqi llgiln malum metodiki iisul saxlamlrnrgdr.
III fesilde qeyri-m0syyan inteqrallarrn hesablanmasmda ags& tortibrekurent dlisturlarrn avazina daha semareli ilsul verilrnigdir. M0ayyeninteqralur tatbiqlarina aid goxlu sayda misallann helli verilmig vstoleholarin sarbest iglamasi 090n tapgfiqlar verilrn$dir.
Teqdim etdiyimiz dors vasaitinde verilen tapgnqlar va misallar asas
ders vasaiti olan B.P.Demidovig "Riyazi aralizden mascle ve misallar"kitabrndan gutilrlllrnu5diir,
Mii,ellifler dors vasaitinin el yazmasmr oxuyub dayarli qeydlar eden
amakdaqlarrmrza ve komyuter qrafikasna gdra Hacryeva Gilnay Fiketqlzma 0z minnotdarklmr bildirhlar.
* I.ilT*"*t
g r. niv,Lzt ^lxl-lzhr ases rm,lvt$r,nnr
g6ry2t6 lorrm 6laceq bezi dtlstrnlarm isbatl ila magtul olaq.
l. r*z*...*n="(Tt).
Eellt n =r onwaa r = l(t; t)
= t oldu[u aqkardu.
n = r olduqda r + 2 +... + t = !(tj-!) berabarliyinin doFuluf,unu qebul edak'
z = t + I ugfln beratrerliyin dolnr oldulunu isbar erneli,yani
r+z+"'+1'+9=Q!f;12
r + 2 +... + r + r + r = i(t-:!.
1r. r1= !G:{?0:D - (t + rXt + z)
isbat olundu
2, tz + 22 + ...+ n2= {z + t}zz + t) z - in istanilan natral qiymetinde
berabarliyin do$uhr$unu isbat edin.
flelll bW ugthr riyazi induksiya 0sulundan istifada edak' n=lolduqda berabatliyin do!rutulu a.gkrdr. n ugun yux,ndakl beraberliyi
qabul edak. r+l oqlfn isbat edek
12 +22 +"''t2*('*'F =4l+al])+('+rf =
- {z+ tXzz + r)+e(r+tP -Q+rhnz +n+a,,+al -66
-b +thr2 +tr+ al- (z+rXz+zX2z+3).
3. 13 +21 +...+ n1 -- (t +z+...+"f .
Hetti. n=tolduqda t3 = (44-,), )' lz=r= 12 = I
z=t olduqda 13 +23 +...+ t3 =(t+z+...+tf qabul edek'
z=t+l olduqda isbat edak ki,
t3 +23 +...+(z+tf =(t+z+...+("+tf
l3 +23 +..,+ 13 +(r+rf =(r+z+...+rf +(r+rf = t2(t * rf *6 *1y =
G * tYG' * * * q\- (t + rf (t * zf
isbat olundu1. l+2+22 +--.+2n-l =2n -1.Halti a=t olduqda t=t
a=t olduqda l+2+22 +.--+2k-r =2* -l qebul edok.
n=t+l olduqda isbat edek ki,l+2+22 +,..+2k = 2k+l -,
1+2+22 +...+2k-1 +21 =2k -l+2* =2.21 -1=21+t -1isbar olundu
L Bemulli baraharsidiyini isbal edin:(t +.r1[t + 12].{t + r,)> I +.r1 +.r2 +... + r,,
burads x1,.r2,..Jn eyni igareli olub, -1-dan b61'iikdtr.
Halll n=12 olduqda benbersizliyinin do$ulu[u atkardr. ,, ugtln
bcrabersizliyin doflplulunu qabul edak z + I iigon isbat edek(t + r, [t +.r, ].(l + r,[ + x,*, )> (t + r, + r, +... + rn[1 + r,*,)= I + rr + r: +
,..+ x, + r,*, +(r, + .r2 +...+ rrlr,*, >1+ rl + 12 +... + r, +rn+l'burada
(.r, + 12 + ..' + r, !,*, > o
barabersizliyinden istifada edilmi+dir.
6 (+rf >r+ru.trIalti Bundan awelki misalda olan barabarsidikde
'l='2=" =xn='g6tt[seh isbat a$kar olur. Yeni
(t+ 11[t + r2).{t+.r n)>l+ 4 + t2 +...+ r,(t + rlt+ r)..(+ r)> I +r+, +... +,(t+rf>t+'rr
isbat olundu
7. r>l olduqda
,.Iz+t)'zl', I
bsrabarsizliyini isbat edin.
Helll 1,2,...,n adadlerinin ededi ortast ile hendasi ortast arastnda olan
1+2+.'.+n r n11t-n
munasibatdan istifada edek.| .n(n+l) r4r.. ,*l r4lr,.n22
berabersizliyin har tarafini n - ci daraceden qtiweta )'tikseldsk Onda
|,,*r)",r\2)olduf,u alurr.
E. 2,,4....(2nl>lb +llr (4 > l)
Halli n=2 oldtsqdaztqt>l(2++f4E>36
z=* olduqda 214t.... (2r)> (r + l)lr (2)
z=*+l olduqdaz'.+....(z* +zl>l(t + zlf *l
(zt, + zl> (t +tl(2) harabarsizliyinin har tarofmi (zt + zf-ya vuraq:
2t 4r...-(2k)t(2k +2)!> (* +r!f (2i + 2)l
2t4t ....(zk)t(zh +2)r, [(l + r)lf (l +l)l
2t 4\...'(2k)t(21+2)r' (r+r)f *t
^ I 3 2n-l Ie .-.....-<-- borabersizliyini isbat edin.2 4 2n "lzn+l
1 3 Zn-lHalli t,=ri- -igare edek. Bu ifadenin kvadratrna baxaq:
^ z tz 32 (2, -tf . tz . t2 .(zn-tY -." =7'7 .. n#'22-t q2-r', (2|,Y_r
t 32 s2 (zn-tf - 1:Q:mrI G:m;, G{Fl, E,-'Xr,-;i- r,+t
Buradan.1
" 2n+l
Har tarafden kvadrat kdk alsaq
o . --L-''- J2^+toldu[unu alartq.
fi. t*L*l*...* L, n b>2).^12 J3 Jn
Hallillllr-1+:+ :+-..+:>n.-=alnJ2 J3 'ln 'ln
11. n"*t > (n+tf (n>3).
IIeltL n=3 oldsqda 34>43=81 >64 Itkardlr. ,?=t olduqda
tr*l > (k+lr qebul etsek.
r=t+l Ugon oldugunu isbat etmeli:(t +t)i*2 , ft r:f -1
(r +ryr*z - r**' (t-1{.', 11 * 1y C4),1-1 = G.'I= = [qOI.' =
= l',::t-,,)*.' = (*.,.i)"', (r+ zf *,
I /" ! n
"' F'[-I,'-,ltE,'*" (0<r1 <z;'t=or'2"')
IIaIElsin(ry + 12 +... + xr ) 3 sin r1 + sin x2 +.,. + sin .r,
a=t olduqda $in:1lssinx1z=2 olduqda lsin(rl +12)ssinrl +sinr2
lsin(r1 +:2 ) = lsin 't1 cos t2 + cos.tt sin ,21 s lsin 11 cc .x2l+ lcos .r1 sin r2i < lsin .r1l+ lrin r2 |
olduqda qebul edak ki, n=k
lsir (ry +,r2 +... + x;) ( sin x1 + sin .r2 +'..+ sin 11
doSudur.z=f,+l olduqda
lsin(r1 +.r2 +... +.rp*r) ( sin I + sin:2 +... + sin rp*1
olduqda isbat edak:
lsin(r1 + 12 +... + xt +.tp*t ) =
= isin(r1 +.r2 +...+rt)cos.!r+1 +ccs("1 +x2 + ... + 11)sin rlr 1l <
s lsin(x1 +x2 +.-.+.r1)+lsinx1*11ssin.r1 +sin-r2 + ..+sinr; +sinr1*1
13. sn-- .,as! + otagf,* *ngL *.-.* rrrrl| cemini upm'
E ll*Iu"=sctE#
igare edak Malurndur ki,
*"tO-*"tg,=ar"gi;.Bumda
ll- 2n-l' ' 2n+l
qebul etsokll
U,= tctCr;- odC 2n+t
alm.r. Onda ( t\( t 1\S"=ltctst- octc|
)+lnctc;- sc,8;f
Itl\LtI+...+locts;;- octcT; r)=
oc,cr- ecq 2n+t= 4-
racrrnt
s= lim s-= 11ro(z-rnr*-)-\=t.- "--r-
" n-+.[4 "2rr+l) 4
71. 12 <2 borabenizlilni Od.yoo bUtotr r yur@n va aqaEt serheddhi
taptn,Eelll
r < tJioldulundan doqiq aga& doqiq yuxan sshdi -Ji olur.
I5, Tutsq ki {-r) goxlu[u xe{r} ededlerina eks olm adadlar
goxlugrdur. lsbar edin ki,a) inr {- x} = -sup{.r};b) sup{- r} = - iDf {r}.Ealll e)1. eser k) goxlulu yuxand& mehduddurs4 {-r} goxfuEu
agaSrdan mahduddur. YeniraM
olmasmdan-x2-M
olmasr aLnu. Buradan aydm ofur ki, sup{.r} olmaslndan inf {-M} olmasl
almr.Tutaq ki,
suP{'}= Y"ba;qa sdzle
x<M'Yani ixtiyari r e {r} va ixtiYari
a>0{lgtin ele x' e (r} var ki,
M'-s<x'<M'do$udur' onda
- x>-M'v,
]t' s-x' < M'+sburadan almr ki
inf [-r)=-v'=-*Pk].b). alda olduEu kimi iof {-r} olmasmdan sup{- r} olmasr almrr. Tutaq ki,
iDf E)= ,t'+' var ki,
m'<r' sm'+e
- ^t - " ,, -r' <-a'
srp{- r}= -21' = -i"f {r}.
yoni r>a*,va>o uglln
Onda -r'<-lrl' ve
Belalikle,
RiyMiI6
T7,
1t
19.
20.
Qoltgnoler:
,filu}Jtya maodu lb lsfu cdtr:
t2 +22 +52 +...+ (2n. 712 =4a4 -t\
t2 _22 +12 _...+(_1),-rz2 =1-g*14,'-]!.2
="-(i,J
21.,,4
23.
21.
25.
26
27,
2t.
29.
l+3+5+...+(2n-l)=n2-q + (a + d\ + (a +2d) + ...+lo{n -t)di=f,fza(n -t')dl, adeR.
1.2 + 2.5 + ...+ n(3n- l) = n2(n + t)
1. 2 + 2. 3 +...+ 4n -l) = 1r(r, - lxz+ 1)
Z. 1 + 3. 2 +... + (r, + l)Zn-l = n' 2n
t-22 +2.32 +...+(n-l)n2 =L,<n2 -l)(3r, + 2)
13 + 33 +...+ (22- l)3 = o2 (2n2 -t)to *Za *...*no = *nb +1)(2n +l\3n2 +3n-l)
f *25 *...*rs =*n2o+t)2(2n2 +2n-1')
TE-1(3t-2X3i+1)n1 _ n(n+2)t
llr1Z*-tyztr+t11Zt+3) 3(2n +1)(2r + 3)31.
n(3rr+l)
k2 _ r{n+L)
31. t sin(a + h) =t=0
E cos(a + h) =t=0
tEr(zk -ry.zk +l) 2(2n+t\
+ I =-{t- ' l. '.v.r:l t(t +l)...(t+n) n[rd (r+l)...(z+n- t).]
. n+l .( r)sm ---r.sml a + -, J
.xsm-2
. n+l ( r\sln ^ x'cosl a+;ri__:_-__:_____1J, x+2m2, eZ
,lsln -
2
*osf,* *og!*...* *ns\ = *n,-!-
(,-il'i){' ;r)=#1
36
37.
l0
It2n-l-Zn+32'12 "' 2n '
2n
I +lr+1ll+...+ r r...r =!a:2t-19u
12.
13.
11.
15.
16
17.
1E.
l, dnE
1rl.r1rlr,*",8r* vrg ?
+...+ 7t8 7 = icrs-. ctsx, x + m E,
meZ.
A;a&dah harubcrsizllhbn rvazi induhslya mclodu ila gAcbrh:
11. Qn)k2zn btY.1 * 1 *...* I rl.
n+l n+2 3n+Z(rA)2 > n' .
(r)2 <f(n+ lx2r+ l)l'\6)
, .\3nOr\4 <tt^l!:Jl
\t)1 * I *...* 1 ,1n+l n+2 2n 2
11 l-- +-+ -..+-<2l! 2! ,n
(, * !)' .l,, * Il'"\ ,r/ \ z')[r* I *...*1)'.2'\ 2 n)
$:L,rtlyJkJn-k+1
(r)'.r."l.1')'\e) \2)
l.r*I']',"'-]t . n(r * !).1
n+l\n)n
52.
53.
ll
3E,
39.
s 2. ARDICILLTQLAR NOZORITYOSi
Tutaq ki, bciyari s>0 adadine qargr ele N adodi @maq olar ki,
.M den sonra galen z - ler ifgiln (z > lv )
la - x"l<e
barabersizliyi odanilir. Onda o adedi
\'r2'--tn,"'ardrcrthPmrn limiti adlsnr.
Axrnncr barabersizliYio -E <Xn <A+ A
kimi da yazoaq olar.adedi a - nun segilmesindan asrhdrr'
Simvolik olaraq a ededinin {r,} rdrcrthf,rrun limiti olmasmt
lin xn =an-+@
kimi yazrlar.<ig". k,) ardlc hfrnrn sonlu a limiti varsg onda bu ardrc hq yrplan
ardrcrllq adlanr'- eg.i ardrcrlh$n limiti yoxdursa ve ya sonsuzluqdursa' onda ardlc lqd"Erlan ardrcrlbq adlanr.-" Ioai .tat"tmgrn limitinin tarifinden istifade edaroh bszi limitlerin
doEnrlugunu gostaEk.
5J. Omumi haddi + = -!- -a baraber olan ardlcrlftlr"n limitinin I
olduEunu isbat edin.hcAt lxtiyart e > 0 adedini gotllrek' Odade qarqr ela N natural
edadini segmek olr ki, nr - nin bundan soma z > 'tV qiymatlerinde
lr--4.' (l)I r+ll
berabersizliyi Odenilsin. Buadan
Lf--1."- I ."-r*lrl=r,1-1.ln+ll z+l e E
Belalikle, iv natural ododininin ,v=l-r qiymatindan sonrakr n -lsr
Ogiln (l) berabarsidiYi Ddenilir:
"=J- olarsa' rv=f -r=ee, iv=p'1oo loo
Onda /v - nin 99 - dan boyrrk qilmetlarinda
[-+1.figarti 6dmilar.
Tutaq ki, nr=97olsun. Ond^ *,=# oldulundan
e7 l- 971 l r l& = er, l,
- 9sl= es,
yonl ->-.Oger z=9t olars4
98 1981 l I r&" = ee' l,
- eel= s9;se, too .
Belalikla,
lim z =1.*+o n +l
ardrcrllqdn ve limiti srfra beraberdir.flalll e>0 adadine qargt
55. Gosterin ki" ,, =(-tl*t (n=t,2,..) ardrc hlr sonsuz kigik
lc,r-'l=1.,,,,1=,v(")I n ln 6
e 0.r 0.01 0.001 0.0001
JV t0 100 1000 10000
56 Umumi heddi ", =2!+ ohn rdrcrlh@ monotooluEunu" 3n-2
araqdnn,flelIL an*1- an ferrqina baxaq. VzerY 090n
- 2(n+t)+3 2n+3 (2n+5X3n-2)-(3n+1X2n+3),an+t-a,=rQ;;i:i- 3^J= (3"_t[3r J,, =
= -13 .o(3n+1)(3n - 2)
Demali, Vz e lV ilglin ar*1 - a, < 0 vo yaxud ar*1 < a, tcrabarsizliyi6enilir. Bu isa o demakdir ki, ardtc Lq azalandr.
57. Umumi hed.di an = J n + 3 - Jn olan atdrcrlh[m monotonluEunu
aratdrm.g"gi an+t
nisbotine baxaq. vn €1v ligtnan
q-.r Jia - J,*r kA* - *- tllr. n*Jr* rIJr*: *,.,6)-%=-Gl-l;-=w-
Jn+t+Ji- Jr*+*Jn+t -".&4 q 1 vs 6usadan aral<d, lkfi\. Demsli, baxllan ardrcdlq ezalandr.
ttrt
J& Umumi heddi a, = --I- ohn ardrcrlhfrn on b0y0k heddiniz' + 100
taprn-Ila[i Verilmig ardrcrlhsn monotonlutunu aragdraq:
(r+ l) n n2 +n-100o"'r -'n =
G, rf *6- 7. lo0 = - ;t;t - lotff;loo)
Aydrndr ki, n=\2,...,9 olduqda a,*p > a,, n > l0 olduqds 'tsa ar",1< oo
berabetsizliyi dof,rudur. Deraeli, baxrlan ardrcrlhq n=10 nOmrasindan
ballaya.aq azalandr. Diger tsrefdan o1<a2<...<q<ar6 barabersidiyi
doSudur. Belalikla, o, = -L b*rl* .rdrcrllfrn an boy0k hallidir.
59. Tutaq k! {a,) ve {6n) azalan ardrcrlhqlardr ve VreN ilgiirl
ao>0, b,>o do$udur. lsbat edin ki, {a, -6,} ardrcrlh[r azalandu.
Holli VneN 09lln arrl <an ve bo11<b, berabersizliklari do!rudur.
Vr€,v llgun an>o, bn>0 oldutundano,a1 bna1< an' bn,,1 <on' b,
alanq. Bu isa o demskdir ki, {an '6n} ardrcrlhfr azalandlr.
60. Isbar edin ki, trmumi hsddi * = #:olan ardrcrlhq mahdurldur.
Eaa o ^
= fi =, # = i]# = ;(, - ;-- ) = ;. i ;-oldu[undan, aydmdtr ki, vrEN 09un c,>]. Demsli, {4,,} ardrc h[l
agalrdan mohduddu r. d, - *t ferqina baxaq.
n+l ,t+2 3a, - ar+t= Tii-
-2n +t= (2n -l)(2n + t)
oldu$undan an+t<qn alartq. Bu iss o demakdir ki, verilan ardrctllq
azalandrr.Ona Eote ar=z bu ardrcrllrprn en bdyUk hsddi olacaqdr, yeni
vr e 1v Ug{in ara2 barzbrsizliyi Odanilir. Demeli, 1a,} ardrcrlltf,t
>0
l1
yuxandan mehduddur. Belelikla {a"} ardrcrlh$ hem aqa!t'|"n, hem da
yuxandan mehduddur.
6L lsbat edin ki, umumi haddi a, = t2 olan ardrctll-tq qeyri-
mahduddur.
Halli Aydndr ki, Vn e 1v 09iln z2 > I . Demeli, (4, ) ardrcrlhf,r
aga$rdan mahduddur. Onun yuxandan mehdud olmadltutl gost rek' tutaq
ki, verilmig ardrcrllrq yuxandan mshduddur, onda ele l>0 ededi vr ki,
vn € N flg{in n2 < ; barabersizliyi odmir. J7 sdadindan boyiik ze natural
ededini gOttlrak. irg olaraq, mesalen, ,j=1J71+1 86t0rmok olar. Onda
,,r=,fr=[J?l*if ,l^tel >{^teY =t alanq. Alman ziddiyyat onu
gdstarir ki, baxrlan ardrc lq yuxandan mahdud ola biknez
62. Teifd;rr istifada ederek, o, =I11 t4rr1hlmn l-a beraber
oldugunu gOstarin,
Holli Glstetak ki, bu Va>0 edodina gOre elo ao(E)€N n0mresi
tapmaq olar ki, n >zs(a) olduqda
l4i-l-J."ln I
berabarsizliyi 0denilir.ln+l -l I Il---ll<6 () -<e Q n>'ln I n €
zo(e) olamq, 1-a- Uoyut t", transr natural ade4 nr"rrf"a atrl=11]*rgot0ra bilerik. Onda aydrndr ki, zo(a)dan Hiy[k her bir natural z adadi
UgUn
berabanizliyi tidenilecakdir. Bu iss tarifa g6re ,lim
Ill=1 olrnasr
dernekdir.
6r, [m +=o (t>o sonlu adoddir) olduf,unu isbst edin.n--+o nl
IIatIi G0sterak ki, Ve>0 ededine g6ra ele z6(a) ndmrasi var ki,
n > zr6(e) gertini odeyen biltiio zJar 09iin '#-rj=;."
bsrabarsizliyi
l5
z+l I I I I 1; -1=;',0@\=fl;'l="
odanir. I . r berabtrsizliyitri n-e nezs?en hall asak , '[])] *'n'r - \8,/n
|-..1-],ot"l=|[1Jt
I+1goturselq onda n> z6(a) gertini odeyen butth nJsr 09ltn
L'-lI I I . I _=,7'[=7--;.r-n-,, r\r" llrr)i l-,1 lrl)t I
ll("./ l''l l\a/ |tl lJ \ )
alarq. Bu ise t > 0 sonlu adod ofdu9aa ,rim-.)
= o olduEunu gostorir'
6t isbat edin ki, .rn = (-1)' ardrcrlhsmrn limiti yoxdur'
Holll Olsmi farz edak. fttaq ki, ardrcrlhf,rn limiti var ve a'ya
' ' ,*l) etrafrna baxaq. Bu etrafin uzunlupubarabardir. a noqtasinin l"-lJ', ,l2-"0 ULU"nait. ArdrcrthErn bildln hedlari y'lnrz -l va l{an ibarat3'oldufundan gottlrdoyutrt{tz atraf eyni zamaoda 'l v€. I noqtelarini oztlnde
saxhla bilmaz. Mujyyanlik ugtln tutaq ki, 1 n0qtesi bu atafda daxil deyil'
l-e barabar olan hedler isa sonsuz saydadr.
Belelikla, baxrlan etrafdan kanarda verilmig ardrcrllrlm hadlarinin sayt
sonsuzdur. Bu iss onu g6sterir ki, a adedi baxrlan ardlcrlLrpn limiti ola
bilrnez. a edsdinin ixtiyari olmasrndan isa almq ki' verilmil arfictllttnlimiti yoxdut.
65' G0starin ki, limumi haddi ;, = zr ('t > 0 sonlu ededdir) olan
ardlcrlhq boYlYondir.
HdlE Btzbilirik ki, 0mumi heddi ,,= j 1* to sonlu edaddh) olaa
ardrcrlhlm sonsuz kigilandir. Onda yr$rlan ardlcillt$n limitinin
yegmeliyioe esasan t>0 sonlu edad olduqda {nr} sonsuz boyuyen
olacaqdr, ba$qa sozle lim nt=+<o.n'+@
66 isbd edin ki, tlmumi haddi r, =C!'*l olan rdrc hq sonsuz"n
kigilondir.
l6
N3an
\
Halll Y e > 0 g6t0rek. G6sterek ki, bu e -na gOra ele zo (6) n0mrosi va
ki, a>261e; gertini Odayen b0ttln z-ler 09lln lC{.tl=t." berabersizliyiI n ln
odenilir: 1<a o ,r!. 4(e) olaraq ,0G)=[f].1 goturse! on&
z > zs(6) Sartini od,oyen biit[n z Jar llglln
,,-r"+tl r r l lf-=;',ro(4=lq't="
Lal E
b.,rabersizliyi dogu olr. Bu isa verilmig ardrcrlhpn sonsuz kigilenoHupunu gostarir.
67. lsbat edin ki x, =lg{lgn) olan ardrcrllq sonsuz b6yliyandir.Helli Ve >0 got{irek Goserak kf U adadine g6rJ ela ,ri0(6) ndmresi
vr ki z > 4(e) gertini odayen bltttln z-lor 090alg(lgz) > e
berabsrsizliyi do$udur. Bu berabersizliyi a-a nezaren hell essk,
z>lolo"
arnq. ,s1r1=[ror0"]+r eOtllrek. onda Vz>rs(a) [g0n lg(lgz)> a
barabarsizlil Odon ilacskdh. Dogudan da
rurea, rd{[r0,,, ].
r)), r{rr,0,,, ) =,.\
Belelikla, Va > 0 odai iigiln ele ,rg(s) ndmrssi taprldr k! bu n6mradensonra galan bllflln zJar 89ttn tg(lgr)>e berabarsizlil Odenildi. Bu iseUmumi heddi lg0grr) (n>2) olan ardrcrlhfm sonsuz bdyllyen oldulunug6storir.
6t lsbat edin ki, t, = o{-r)' ardrcllft$ qeyri-mehduddur, mcaq sonsuzb0yllyan deyildir.
HcM Ye >0 vo ,o =rh(a)=2(t6l+l) ctt natual ededini got{lrak. Ondaxq =2(ltl+t)>2c >a barabenidiyi Manilacckdir. Bu isa onu gOsterh ki,baxrlan ardrcrllq qeyri-mehduddur. Diger tonfdon, n=zk-t (kev) takadedhri 090n r,=f olr. tstenilan [o;a] (a>0 sonlu adeddir) pargasrna
nbanlan ardrcrlftprn sonsuz sayda heddi dxil olacaqdu. Bu ise onu g0sterirki, bax an ardrcrlhq sonsuz b6y[y-en dSy4._ -f
17
6s. isbat edin ki. f*1, Fi=t ardrcrlhfr qeyri-
mehduddur, ancaq sonsuz bOy0yan deyildir'r G-l
ura l,,l = lrin4{llf ardrcrlhgI n'+4n )
mehduddur ll.""f-1.,.l( z'++a i J
Diger terefden asanhqla gOstarmsk olar ki, ,, = *, z e tr' rdrcrlllt
sonsuz kigilandir. Onda sonsuz, kigilenle mahdud ardrcrlhf,m hasilinin
sonsuz kigilan oLnasrna gor5' {-!,t,#12 i ardlcrlhlr da sonsuz
[z+1 nt +4n )
kigilm olacaqdr, Yeni
ra | "iooW =0.n->an*l / +4n
70. p -nafiial edad olduqda limiti taptn :
.. lP +2P +...+ nPlrm
---=---
n--r@ 1P+r
Halll an=1P +2p t...+nP, 6,=zP+l gdt0rek' Aydrndr ki, $tohs
teorominin birinci ve ikinci gertlari 6datu' bTfi=;i#-;Anisbatinin limitini tapaq Nyuton Binomu dusturuna g6re
(n+l)P - ,.^ nP + P'nP-l +...+'l16 r"Tr' =lim##l;*r,r-t -np+t ;;-np+t +1p+l)nP '...+t- nP+l
- ,.^ nP + p.nP-r +...+r= ,* t*';n *
r,l = , .=;T. (p-l)r, '--l -#lir* j*...-,1 P+t'
71, in t(a>r) limitini tapm.n--ru an
Halll an =r2 , b, = on gottirak' $tolts teoreminin birinci va ikinci
gertlari Odanmesi aydrndr. Ug0nco gertin Odenilmesini yoxlayaq:
rim on.t-dn =n^@*,1)'-!' =ti'r-2nil=
l - li-2"1'=i-\b,-,-b, ilii o,rt -on ,*a"\a-l) s-|n)* on
1E
2..n l_. I=- |lm _+_l!m_.a -ln--r@ a, a_l*+a an
gtolsts terremini-{ nisbatini ratbiq edek. Ondaad
Ii,, lL= [m n+!-n = l ti. l=nn"+a gn n'+a on'l -an a-l;;@an -'
Belalikla, tim a,n*t-?n
=o alanq. Demali, firn 4=0.n-+@ on+l - Dn n-+a a!
72, x,=;+: Q=t2,...) ardrcrlhgmm daqiq agaSr ve yu@n.,i z + I00
serhodlerini tapm.HaAi xn*1-,r, ferqine baxaq. Vre r'g Ug0n
n+t r_ _ (z + lX.,[ + 100) _ dl6Jl + loo) _.t-,.-:-=--Jz+ I + lfi) Jz + tOO (.fi1J + rOOX.fi + tOO)
JiJi\.1,i - Jit +too- " '' ''->0.(rG-i-i + t00[.l!+ loo;
Demali verilen ardlclllq artandrr ya yuxardan qeyri-mehduddur. OndaVeyergtrass teoremina gdre
,,P1,r1=ruPrl =
' |=n a IJz+IOOJDigar terafden, bu ardlcrllq artanolduqda olacaqdu. Demeli,
,r{,,}=i,rf , }= ,1,.d n [J;+lool tol
7J. Umumi hoddi r,=(-l)'+tt!1 om {r,} ardrcrllfr $90n
inf(xr), sup(rn), |!qn :, va li-nr rr-i tapm.n n ,-)@ *)a
Halli n =2k (r = 1,2.... ) olduqda xil_t = - rh;n =2tt -t (h=1,2,...) olduqda ,21=)-a1
alt ardrclllgml alanq. Aydndu ki, {x1} ardlcrlh[l azalan y5 a$^Edan Iilo rnehduddur. Onda Veyergrrass teoremine g0re bu ardrcrlftlrn daqiqagall ssrhaddi var va
lim u =ro.
'+- Jz +100oldutundan kigik qiymatini z=l
inf{r21} = lim r21 = r,ln [r. ]l= I 'n n"r@ t1_+@\ L4
'Diger tarefd.n, (r21) ardrcrlh[r azalan olduf,undan 'r:1 * ardrctlhgm
an boylrk heddidir. Demeli,
sT 1x211=] ',2
'2*-t =-;= (t=Le.'.) ardrcrlh[r artatr va yuxaxrdan mehdud&u
(mesabn, OJa). D€mati Veysgtrass teoremina goro
T,z*-r = r!lL,zr-, = -gJ- ;-r) = t
xr =-l bu ardrolhBrn en kigik haddi olacaqdu, yeni in{{:2,-1} =-1 on&
"o{"} =.{;'o)= J; inrk,} = min(r,-r)= r'
0 va I limit noqtaleridir. Aydrndf kt, bafCa limit noqtaleri yoxdur'
Demelilim .r, =0, lim r, = l.n-+@ '--'o
71. r,=t+n.sin!! ardrc hlr ueun r1r{:"}, srp{rr}, ,,!!!0
r, ve
i6.x,-iapm.
Halli n=4n, n=ln-1, n=4m'2, n=4m-3 (n=1'2"") hallanna
ba:<aq:
n=4m olduqda, ra. =r +4m'sin2nx=l;
n=4m-r olduqda, ra.-1=1+(4r,-l)'shgI;E =r- (4n -1')=2 - 4n;
n=4m-2 olduqda, rlr-2 = l+ (4n - 2) ' siag!:2ls =l'
n=4m-3 olduqdg -r4r-3 = l+ (+n - 31' si/@J)L =4t-2'
lx4^-11, meN ardrc hfr azalan ve agalrdan qeyrimehduddur' Onda
Veyergtrass teoremina g6rsi'f{'a,-t) = lin (2-4n)=--a '
')@
{xa.-3) ardrcrlhlr artan ve yuxandan qeyi-mehduddur' Onda
Veyergtass teoremine g0rs
sup{x4--3} = lim (4ar-2)=+o.,i ,t-r@
Diger tarafden" lim xq^= Iim r4-_2 =1. Baxrlan ardrcrltrq hemn -+6 fi-+6
yuxandan, ham dc aga$-rdm qeyri-mohdud oldu$undan llgq r, =..o,
lim r, =1q olacaqdu.
Digar terfden, inf(r4.-1)=-.o, sup{r4r_r} = ro oldu&mdan
inf (xr) = --<o, up{rr} = r< olacsqdg. tn
n,
75. xn =w'! ardrcrlhgrnrn agaf,r va yuxan limitlarini taprn.
HcIIL n=3k, n=3k-tt n=3k-Z (k =\2,...) hallanna baxaq:
n=3k olduqda, x31 = cos3r 2'liz
= cosu 2tx =l;z=3[-l olduqdq
x11-, =p,,"rl) 431:t)r =**-r(2*o -!)="**-r ? -1- r 13r-t.3 ( 3) 3 \zt
,t =3t-2 olduqd4
4 s _, = *3 k - 2 43k : 2) r = ** a (x" -
g) = (- ;)t
"olar. Indi ise xususi timitleri tapaq
7 1 13/<_llim r31 =1, lim 131-1 = lin l-: I =0,
f --r@ i-io L-+@\ Z)
*u'--'r*-, =.{ - I)'r -'
= o .
Belelikle aLnq U,[E r, = o, lim ,r, = 1'
n-ro n-+@
76 Kogi meyrmd*n istifade "a"ot
,, =E,al ardrcrlhf,mrn yrFlan
oldutuau g0starin.
Halti p,*, - "n1., ,rrltlpn*n-a"1=O *
(,+ 2x + -.+ (,+ pr.
2l
Riyazi induksiya rnetodrmun kdmayi ile isbat etmak olar ki, vzefi oQiin
a> 4berabarsizliyi do$udur. onda vne iv ueun ,1' j berabarsizliyi
dotsu olar. Bu berabersizlikdan ve vt >1 natural adedi ttgiin do[ru olan-1111
k2 - h(* -rl k-t h
barabersizliyindan istifada etssl alanq:44
lon* p - onl. G;f + - +
7- .-t r.
.{i-*).(*-#). .['; #)]=i; #,)':1<e barabsrsizlivinden r>1 ahnq'n€
77. Kogi meyanndan istifade etuiekle '" = ir+* ardrcrlhlrnm dafrlan
olduf,unu gOstarin.- - ''iUi- Bu msqssdle Kogi meyarrna esas€n {',,} ardrctllt[mm
fundamental olmadrEml gostermek kifayetdir' V,rp - ',1, ifadesini
qiymetlendirek. vn € vo p>l ededi 09{ln
, r | ._)__...*|:r-__I_:Vn+ c - tnt=
-. ffi-'-- fiV - .1; *
berabarsizliYini alanq.
Bu barabarsizlikds xOsusi halda, P =', " = +
goflhsek,
vr,_,,1.h=Lari="olar. Demali, {r,) fundamenal ardrcrlhq deyildir' Ona gOre ds verilen
ardrcrllq yrlrln deyildir, yeni daprlandrr'
,r. ,. n7 -+2n2-+1n+
4 .
n--+a 4nJ +3nz +2n +l
IlallL
,f +2n2 +3n+4lm - - = ttmn-+o 4n5 +3n' +2n+l n-+'o
"'l'.': i.'r) -,,'F++.Hr
7s. nn E:.,-.jzE+32+4
HeIIL
-4 ^- 1n -llrm --= lrmn--+a rlnt n3r*4 n*
=lim\(.tt-&,-3 -,["'; 4" i\.[,2 . s,,. 3.
= rin:l-: =!=!=z'-'rl'.ili . {*a,* ) 1*' 2
at. ri,n 31'-t .
n'+@5n2 +2n
HoM
ri^ 312-r= ,,, {-=1J,,4',5n2 +2n ,-. n2lS * 4
)
,'(;]r\= rim---!-l-=3.''- *l('.).i)
3-0 3
5+0 5
;-,lrrz *n^)=
= rir, :9:= rir',:
''" l,,ot!. *)ft* f,,,t*!* )) '*
27
u. r, (^tF-t"-t JF-r,i)
t2- tirr -L.n'-,'-- .,112 *1
HaAlim -4= ,t, ---ft
-=-l=-r.n-l,-a.tr2 *1 ,--- r-ll * 1 I
\n'| _1n+2
tJ. lim 'n)aa 3-7n
fle,U
ti^ l-|n" = t. 7-" +72 -o j 4?
- 49 .,'-+-+- 7-7n "ii 1'7o -1 0-l
Ilaradakr, lim 7-' = 7-' =0.,-)-[<o
t1. tu 2+4+6+...+?nz-+lol +3+5 +...+(2n+l)
Haai 4?.-
2+ 4+ 6+...+2n .. ,,5;l; * -*tz*r'l =,l%Gr. I {, - D
=
= lim ':= liltr +=1.n-+lan11 ,-*r*;
soJ. rlu ---
^ ,i ..
n-+)olz +2' +3' +...+ nL
HeW1
lim ---=--a-++ol' +2' +3' + ...+ n'
t5. lim -
= limn-+ta t{n + lX2n + l)
m i-(,-fir;;).
6
6s
21
=,i1";o;#;o=
,.6= lrm
---------------:= = J,
z-++"o 11a 112 + L)
nn
Halli
(,-.F -s"\,..[7.s,)n+
_.-5-55tlm =-=-_."-l+-[+l l+l 2
ln
. 41*E!) l+h-Ili- '+h' = tim =4 = lin =I =1.n--sa3n + 2lnn ,* ne*z!r, "-- 3 *2!n 3
nnE8. Limiti hesablaym:
.. l+o+o2 +..-+a'rdE-
E +al+b+bt +...+ bn
.. l+ a+ i + -..+ onlrm-=n+@l+b+b' + -..+ bn
ri- ["-,,f'z+sr]= ri.,--+o\ ) "-+-
-. -5n= lun:=n'- n + tln2 +5n
t7. im n+ lnnn-+-3n + 2lt n
HeAi
U.r lal.t).
trIaIIi Kesrin suret ve mrxrscine handesi silsilenin cemi dtlsturuoutatbiq edek. Onda
1- l*ll' l-a: =
,-+@ | - b'+t
l-,
_,. r-r. 1-,lT-r"-] _r-a.,'1it-a 1- lim Dn+l 1-4'
se n-[-L*l*...*--!---.l."'' i\{t.z' 2.3 ' "' ' ,(r*D.l 'Halll Bu camrn ,, - ci hoddi
,, = I :1- I"' tdn+l) n n+t
oHuf,undan, hadlari
L', =t-!.u. = ! -L u. = 1- I-2 3'"r-3 4""geklinda yaznaq olar. Onda
s[,,r. *. . #D] = J*[, - l).(; - i).
90.
IIeA
Onda
*fl-1.\*...*fl-'')= ri,lr- -l-l=r.-\: +l "''tn z+ll i;-\ n+t)
,*f ",.4- -0-U-l limitini hesablavrn.n-+oln' n- n )Melum diisturdan istifada edek
- ,; +22 +...+{n-tl=k-'Y,,-0 .
*'[4. 4. . fr .,'r I = :*tr*fj =i;g]('-il'- i)=i"-{n' n' n- ) '-ll' t'
-ff=o oldufunu isbat edin'
I/ali Burada Nyuton binomi dilsturundatr istifada etsek
n n - 2n -2;=Gf=;;@P;"=64=;alanq. Buradan
ti- 2 =0
lr-+o rf - Iolduf,u aqkardr.
n )\*=o (a > r) oldulunu isbat edin'
IIallL Tu,rrqki, z tam adeddir va z > I olsun' Onda
o.t-=ra=l-L'l- =l:rl""' ""- o, l4o, ) L* I
l=d7>t. Buradan
o'fr=*fy=2n 2
'fr:m:f =Grxrrrt
yaznaq olar. n + @ olduqds .,;* -;.
= t Beleliklo' z --+ oolduqda
(#)' -, oldulunu alnq.
26
a-rf +...+(a-tf
93. s.n" =o (lql.t) otouEunu isbat edin.
Hallt mn= o =n. t=!' [,]' '"
q
Lcl
irars edak. z -+ co olduqda l -+ 0 oldu[u agkardr.
94. bmali =1(4 > 0) oldugunu isbat edin.
Halli a =l olduqda isbat aydmdr. Tutaq ki, a > I olsun, onda <,/i > tolar.
a = tl + ({7 - DI" = t *,,$li - l*
Buradan
, = [r *(v; - rf, = r * fu; -r)*'{";i(t; -rf .*.... 1v; -ry, d,r; t)ka
- rI.
kn.f.* vo ya ,t;-r.ffi.Barabersizlikda a-+< limita kegsak -L= 0 oldulundann-l
[m {i =1.
*'4': i 1t1; - rl *... * (4G - t'l,,ftt; - t).
Buradan a>z ft/" - r) ,trn".
{i -t.9.r,h
z-+o olduqda 47-t=0. Onda
)'-.._oG=r.
95. \n 4!; =l oldu[unu isbat edin.
IIaIli Melum beraberlikdon istifade edek
Qohqrrtolo,:
Asa!frah misallan lell edin:
96. [,, lqm'.n-r@ n' +l
Cavab:0.
Cavab: 0.
Cawb: 0.
cavab: f .3
cavab: ! -2
cavab:1.2
Cavab:4j
fis. wz!r.=0.
105. lim log"
=o (o rt).
34 (61). tii:+=o.
36/66. lrmL=0.' '-.Vn!
2t
97. rm_(6d-#).
u)y.rye.9ffiroo. ri.ll*4*...*+)
,'+@\n'. n' n' )
lot. rin [-?* 1-...* (-'r"'r-+-ln n tr n
rm' ru'[4*4-" * (z';r)'']'
a-+-ln' o' n- lro3. u^f!+1+4.....*Z:1. cevab:3.
"1ei.2 2' 2' zn ,/
M. E^l7.ttr.8Jr.-.]W). caYab:2.
Asafidah barabarliklari isbat edin:
-, jdi; T)='.18, ,,=(1+1)' A=\2-..) ardrcrllrpr monoton artrr ve yuxarrdao
mahou4 r.=(r*1)'.'{r=r:,") ardrcrlft$ monoton az^lan 39a$d6n
mehduddur. Bu ardrcrllrqlrrn eyni limita malik olduf,unu g6starin:
ri," [r * 1]' = ri-f r*ll'-'=,.,+,\ ,, ) ,16\ n )
ri.(r*r+l*1*...*!l=""--\ 1 3t i.)
3)
4)
roo. ',* [r - !)' = e oldulunu bilerek
oldulunu
r10.
11L
112.
113.
isbat edin.
Onuai haddl aga} dah hfunl olan o*alhqlonttolmw icbol cdhll) z-l
n2)n' xn= "7----:.ln+l
2n'n zn+l
'r=lWlIsbat edin kl ay|4ah ardrcrlbqlar ozthn&r.l\ 2n+ 3. xn=i;:,2)
n' +2n+ 4
3)l ,n- F-'ln' +1
4) t, =log1 n
2
5) un=2-nIsbot editt kL osa! dah adrcrllqlot ,ronotoa deyMit'l) x, = (-1),,l I nrrx' = -sn 2
3) rn =2c;osnr4) x, = nwn'tIsbd edh ki, a7a&tlab adw IhqIfi ,ronoton deyildtt,t) ,,=4,r_4
2) n2 - 4n+32n -3
29
1U.
115.
tI6.
isbal edh kl, atafudah atfuctllqlc ,nfrawn ndmndanbasbyaraq aulonfur.l) ,3
'"=72'l - -.r.+l_1r-lASapdab adrcrlhqbnn rra otonlugutu oagdtmtr)
2)
3)
n2 +7
n
x,=3J)a-1;1n=@t
Jir' =i6o + n
r ooo,,
nl
100
n
xn =n2 -5n + |
2'n
,n =r2 -gn - loo
Cavab: artandrr.
Cavab: azalandr.
Cavab: artmayandtr.
Cavab: x1g6 = I
Cavab:
10001000xrooo = -]ffi! .
Cavab: 116 = 20 .
Cavab:xz=4=-5.
ICava-b: r, = --'9Cavab:x4=x5=-12o.
1)
2)
3)
4)
s)
6)
7)
1)
2)
3)
4)
Aga(rdah a rcr tql&nrt b0! k haddinl taprrl2n+3 cavab: ," =2.xn =:--;Jn-q
nl
"=l;* Cavab: xr6=-:-.
n2 Cavab: x,=2.xr=j " 8
,n=6n-n2-5 Cavab: 'x3 =4'
r, = "lon-n7-24 Cavab: x5 =e'
- J; Cavab: - = l.'"-s+n ' 6
8)
117, ,lSaPdah ardrctlhqlonn au hiql* haddhl tapn.
30
xn =3nz -lon -14,2xn=-i
t) looo''n - ,rl
11& isbst edh ki, ordrctlhqbr mchduddut:l) ,1+1. x,=iiji2)
3)
4n2 +2 -2nn2 +6n+10
(n + 2)z
119. lsbat edh ki, ordtcrtt qtot ,rohduddut:l) an = 1_t1" n2
2) xn = ne\'
3) xn =3n + 2-'4) xn = log2(n2 +n)
DA TeriJdan ts$ada cdenk bfr,t cdh:1)Dmt=o
n-+a: n * 2
2) Iim z+2=ln-+an*l
3) .. 3-nZlrD --=-la_+a n +|
I2l. isbot edin kl asaE dah ailwlhqlarm limiti yoxdur:l) rr' xn=c'o6'
2\ x- = (-l)" +1t
1l zzrrz=Sin-
6)
7)
5) '.=2"+ff
Caveb: ;E = 24.
Cavrb: 12 = -22 .
Cavab: r. = -2."8Cavab:
lo0ol000xlooo =
I Oo0!-.
3t
122. Kogt nt yaflnfua irtifada edank aSa[dah adwlhqlarur yr$lon
oldu{unu bba edh:1) xn=ao+orq+anq', haradakr, lal<u 1* =o't'2""1
ve iql< t.,) sinl sin2 sil n
xr= ,+ *+_+/.1'l cosl cos2! co3"!
- --t-+-, - l.Z .
2.3 . --. t(n+ll
4) ',=r*)*I*"'*)'cts.alris 1. + - f @ = 2),...1 barabersizliyindan----- '
'z s1-l n
istifada edmak.
1,I.. lsbot edin ht, aga$dab anlullqlat sonstu Mylwndin1) xn=(t)n.n2) rn=Zfi3) x,={i4) rn=Wzn5) xn=ldlgn)' n>2
121. Monobn vt ,rahdud ardnlhsn ltmittnilr vatu$ haqqadt
ircrrAa, btifab edanh a$a$dak' ardralhqltnn yt$lon oWaga"a
bba, edtttl) 10 ll n+9' r"=1'1'^ 2ni2)
"=('-;l'-i)t-;)3) "=('.;X'.i){'-})
725, tn
1)
2)
4) ,1=J1, ,2='lilTz'..., ,2=
ardrctA$ Agln inf xn
_lxa=l--n
,+l ) nrrrn =-@s- -n1
sr.prrr, lu4 r, va ltn xn-i lo?tn:
Cavab: 0, 1, l, l.
cavat: o, f, o, t.
z+Jii...*Ji,...
3)
5)
xn = GDn (Zn + r)Cavab: Ardrc fuq a$a8rdan ve yuxandm mahdud
7)
E)
e)
deyil; lim .x, =*, iE rn =+-.n)@ n-+&
n+2 - nnx- =
-stt- _ n> 2" n-l 3'
2+(-ll' I\=2-;
,, =cu*'(z*J).
-nnrfri = I +
-cos-" n+l 2
rr=l+2(-l)'+l+3(-1)n-l 2nz
" n+l 3
xn = (1)n n
,n=-42+(-l)nl
x' = '(-l\'I
" n-10,2
l!!q r, va ii r,-i tqrn.n-+@ ar@
,2 znxt_ =
-cos-" n+l 3
,, =(r*1)'.<-0,*"nf
n - TirrI_ =
-Stn- -" n+l 4
r, ={l+2'.1-9'
,(r:U2
Cavab: -JJ, zJ3,J1 J1-T'T.
Crtrb, -;,:,:,3
,'c**: -z),s,-2,,Cwab:0,2,0,2.
Crtab:4,6,4,6.
c*aa: -l,t-1,l.Cavab: - o,+ co,-@, [email protected]: -o, -1,
-@t [email protected]: 0, + o, 0,[email protected]: -5, 1,25, 0,0.
Cavab: - l. l.2'
co*' -[,*a).\ J2)'e +1.Cavab: 0, 1.
Cavab:1,2.
l0)
1l)
t2)
l3)
126
l)
2)
3)
4)
$ 3. TuNKstYAMN LtMiTi
f. Ogor (a,6) arah[lndsn g6tik0lmii5 ele 'n,M
adedlari varsa ki,- n< f(x)< u x e(a,b)
lerti 6donsin, onda deyirler ki' /(r) firnksiyasr (a,6) arahlmda mshdud
fimksiyadr.,; = ir,f fG) edadi /G) tunksivasrnm atatr ssrhaddi'
xe(a,b)
Mo = sup {/G) "drdi 7(x) funksiyasrmn hemin arahqda yuxan sarhaddi
xe(a,b)
d/lrrrlr,. Ms - ns farqi isa finksiyanrn hamin arahqda reqsi adlanrr'
zo. ltoqtada funksianrn timiti Tvraq ki, /(r) firoksiyasr x = [r]coxluEunda tevitr olunmug funksiyadr, a ise limit n0qtasidir' Onda
i-iiv"i rro iiglln ela 6=o(e)>0 ededi tapmaq olar ki 0<l'-41<6
barabersizliyini tideyen butiln x - ler tlglln
l7(r) -,{ < a
baabersizliyi Odenilsin. Onda ; bu firnksiyanrn limiti adlanu va
hn f (x)= ,t
kimi vazrhr.--'-fi.t Ul. r, -+ o,x, * a(n=1,2'..) ardrcrlt[r u9fln (r, e 'r)
)Yi/G)=eberaberliyinin Odanilrnasi ham zsruri, hem da kafi gertdir'
iki gorkomli limit var:
(l)
(2)
l) lim srnr
= l,' r+0 r2) lim(l+r)'=e .
Kosi fuiterivasr' brtiyari e>o adedine qarqr ela d>0 vsr ki"
funt siyun* t yi, oblastndan gotfi0hniig iki x'va x' noqtsleri ilgiin- o.lt'-\<6 va o <lr" -al<6berabersizliklari odanildikde
f('')-rG')'"barabanizl iyi odanilsin.
lo.Birtirallt linitlar. 0<p-i1<d(e) barabersizliyi 09ltn l'{'- (l) < a
berabarsizliyi 0danilirse, ooda A' f(rl funksiyasrmn o nOqtasinde sol
limiti adlanr vaA' = .ljy-".rG)= r(' - o)
kimi yaztlr.
Eyri qayda ila o < la - rl < d(a) b*rabarsizliyi 09un ll' - fl-r) < c
odenilirse, onda ;' 7(x) firnksiyasmm a ndqusfurds sa[ limiti adlanr ve
,t" = ,t;n o|k)= l(o+o)kimi yazrhr.
40. Sorrste limit. lxirysri ,>o ugltn ela d(e)>o var ki,o<p-al<6(a) barabarsidil 6dsnitdikd6 fft)>r bsrabarsizliyi
Odanilsin. Bu limit
Jg"/G)=..kimi yaztlr.
!27. x-+0 olduqda, a;atrdakr berabarlikleri isbat edin-! ( 1'
a)(b) xsinJi=xi *d,;l' q@) h,=fu-l (t,o):t./vl(i) (r+xf =r+w+o(r)', c)@ *"8:=4).
HatE a);g"if =t. b) rim r-'rn.r=l.r-I=,y-#=.x2
v) (t +;)r =t+ra+ "jx2 +...+ro =
=I + -+(",2,+... + r'-tb=1+ * + a(r),burada a(r)= clr +... + r'-r
l$"G)=o'Onds
(t + ''f =t+**61';'.l tl t
c,Ycrya<,
,rr, ,'-x+x2 + '!x" -n hesablayro.r+l x-l
Hallt x-l= y ovademesini aparaq. Onda r-+l olduqda y+0 olar.
Belelikle,
ri.0:zE-02I:..r-Gd:zy-o y
alanq. Burada Bemulli borabersizliyinden (t+xf <t+nx oldu$unu nazare
alsaq, onda
^| + y+I+2! +...+l+n!- n - r^l'n+(2+2v+...+ry)-n -
y->o Y !-'+0 Y
= go, y(r+z+...+ r) =t+z +...+, =fu
+l)y-+o
,rr. ,,o', 0 * ^I --(t * -)" hesablaym.r-+O ,'
Ealli Kasrin suratine Nyuton binomu dtlshmrnu btbiq etsek, alarq:
(t + n",Y =r + nu,c + 4\r') r' r' * *')(t + ml =1 a 2'* * dir') * * * dv')
,. (t*,*I-=(t+-)'=r-+0 u'
,. t *.**f,^{r-l'lnzr2 -t-.*-!ndn- l)t2x2 _
-ruu--r-)0 r'
=L ^h'n2 -,-2 -r2n'**"\z -"tn- 'l2x-+o ,2 2
130. lint:-l (rz vez ndral ededlardir) hesablayrn.r-+l r'- l '
Halll x=l+y ovedamasini apraq. Onda y-+0 olar. Sonsuz
kigilanlarin drlmasl qaydasmdsn istifada eaek, alanq:
tirr, ,'-l
= * (i+v)" -t - r. l+nu+04)):!-n .
t.+t x" -t r*o(t+yf -l y-+o I + r4., + 0p)- I n
131. firr(-A-- n -) (rr, , natural edadlerdir) hesablaym.'-+tll-rD I-x','
Eelll x=l+ y avademesini aparaq. Onda y-r0 olar. Bunu verilanmisalda a3236 afusq, alaflq'
rhl ' - ' l=',-[ , .rE;r;F -, _ G7 r ivlT;ryi@-
------------!--l=,*[- , . , l-r(r*,a*fu;tt'*6'D ] ;t ,.T'r'' ,*Tr')
JO
( m-l n-l\| __ tvI , 2 f m-n=jy.;*z_r;;tn= ,-\. 2'^ 2')Ir2. Limiti hesablayrn,,. 1[[, * r') * f , *
to ) *... * f, *
(' - t)"11
'-+"1\ n) \ nJ \ n ,/.1
rt 1l[, * 9) *l.,* 2o')*... * f, * ('-')')l=,-.zl\ z/ \ n) \ z ))
=;6 lf1, *, *... * r)+ 9(r + z +... * (, - rfl =,+€rt L n l
= 1;. 1f ,,* s. (r *, - tXn - D'l= rin, !L * 9 fi _ 1)l=,. g.n+anL n 2 ) r+anL 2\ n)) 2
'' '2 +...+(zr-tfIJJ. Limiti hesablayrtr lirr' n+@ 22 + +2 + ...+(Zaf.EIaIli Kesrin surat va mexracindaki cemlari tapoq:
(2n -tf = 4n2 -4n*1z=t olduqda l2 =4n2 -4n+l;z=2 oldrqda 32=4-t2 - 4.1+l;z=3 olduqda 52 =4.32 -4-3+1;
a=n olthrqda (2n -tf =+n2 - 4, *1.Teraf- terafr cemlaselq alanq:
t2 +32 +52 +...+(2n-tf =$'*22 *,..*r')-
- +( + z + ... + z) + 1 . n = a. "(' + tLz" + t)
- o.'1" : t), r - At'l - i22 + 42 + ... + (z,nf = a\' * z2 + ... * o') = 4. fu + tye' + 1)
= ],rb + t\2, * t).
Bunlan limitda nazara alsaq, alarq:
.Bqjm+'o=.qff#+ii=#,l3,L Limiti hesablavm -
t3 ++3 +"'+(gn -zP- .
,--- [r + I + 7 +... + (ln - z)f'EaIIi Kasrin suret va mexrocindaki cemlri apaq:
Qn-zf =27n1 - 54n2 +36n-En=l olduqda 71 =27.21 -54.12 +36.2-t;
n=2 olduqda 43 = 2'7 '2t -54'22 +36'2-8i
n=3 olrluqda 13 =27 '33 -54'32 +36 3-8;
z=32 - 2 olduqtla Qn-Zl=27'nt-54'n2+16n-8'Bunlan taref-tarsfs cemlassk, alartq: ,
t3 +43 +?5 +...rQr-zf =zt\t3 +2r +33 +...+nrl-
- stl2 +22 +12 +...+n2)+x(t+z+t+'"+')-tn=
=!r'h +tf -s,{n+ t[2n + l)+ l8n(z + ])- tr'4
Eyni qayda ila
| + 4 +7 + "'+ (3n -2)=4Poldutundan
[+r+z+...+(:r- 21Y =n1(trtY '
olar. Ahnmrg qiymotleri verilan limitde nazere alsaq' alartq:
21. nz1, +tf -9r(z+ 1f,2,,. r)+tE{n + t)- an
-ft"ffJ'-Ji*J'-'
135' Limiti hesablaYm lim j-=-7- '
'l'2 ' o'
i'r: i(*'ffi *dra. h = rG !Y. i=; +" = hnl-
136. Limiti hesabl"vt l1il!= (z-tam edaddir)'
Halti t!*r-t=y evezlemesini aparaq' Onda v+0 olar' Buradan
1+ 1=(r+ yf +x=(t+ v| -l= t + 'ry
+ di-t= "v'Onda
,. {r+r_r_ 6Z=1r+0 x y;ony n
ols,r.
=]Y,@ftW7.'lg'E=
38
137. Limiti hesabl.r. jgl@l@ (zvaanaturat ededlardir).
.tlaJli Kesrin suretina vahid alave edak ve gxaq, onda
,_r. 4t+- -t+t-l6+frx = t^M-, _1;, @.1r-r0 r xJO r .r-+0 xalanq.
' " dn^'t'l = llm-
r+0 -t
limitini hesablarnaq Ugtln ewelki misalda olduf,u l<imi 4t+u -t= yavezlamesini aparaq. Onda 1+6=(1+yf veya
s =t + ny + ab)-1) pty - x --ryd
vd,' = IUll __:- = -' ya\mt mq
Odur ki,
almq olarq.
limitini analoji,-+0. Buradan
,r=:r\ry.]olaraq ,{*6-t=t avazlamesi ile hesablayaq. Onda
B* =(t + tl -l=l+ar+0(r)- l= nt. r=!.'poldutundan
I. = lim t =i. t_O nl hi
olar. Tapdrtrmrz bu qiymatleri 1-nin ifadesinda nezare alsaq
t=\-r7=!-p=9:-&.mnn
alarq.
13& Limiti hesablaym , = *6+*'{t+ A-t (zr ve rnatural
adedlard ir)..tlafli Limiti hesablamaq ii90n kasrin suratine dlE-i olave edek va
gxaq, onda
, = t -{tr' {iE:4r * *:-4t * *-) -
_ fi,, {l *-({He - l)* 6. GE1 =
rJo X rr0 X
=l,S{i.- 1gryl.:**,ry1== jg'{Tj.
I i^@:J = 1' * 1''
Burada bundan owalki misalm helli qaydasrndan istifade etsek
,. =2. t=o olduluou alanq. Belalikletn'm
I=1,+t.-l *a -ot+ftn''nmlfualmq olarq.
139. Limiti hesablayrn tirl 9l ( z va n nalural edadledir)',+lvr-l
g"7A a = (t + tf' evezlemesini aparoq.Onda r-+l olduqda '+0 ola '
Onda.. 4i-r .. (t+r)'-tl5E;=Hf-f -,=
.. t+ nr + o(t)-t,--ot."*m;
-. nt n
F+0mt m
HIIIL Burffl,- x-l=y ve ya r=l+v avedamesini aparaq' Onda
), + o olar. Belelikle, alartq:
r rtl- lli 1lIt-rt*yyi llr-o*y)j f..lt-tr+r)'I t.. t..t..sw=,-"*j:*=*=*
l4L Limiti hesablaY. ,li1ffi#Helll x=E+r evszlamasini ryarsaq' ,-+o olar' Belalikla,
sl#r=P,H##=mffi=rss##== t-,Y'" + :y,Y ;fu = c'r-" t'
t-+0 ttl
l4tr Limiti hesabrayrn r,, tt - tt -fltt
- *l
tlL 1+sinr-cosr hesablayn.:-+ol + sinF-cosflim
40
flalli Mislt hall edarken milayyan elem€ntar gcvirmelar aParaq vebirinci gdrksmli Iimitdan istifads edsk:
. sinr + 2sin2 I-- $n-t + lI - cos-t I -- ')lim ------------+ = lrm _----- =r-,osinlr+(l-cosA) ,-o
sin a +Zsinz F,2
2sin r
cos r
+ 2 sin 2 r
aa., 2(cm I + sin r)sin r
'22',2= lim = lin
, .-ro 2"; 4[ "65
4 1 2:in2 4 r-+oO*F asin4yia42 2 2 2 2' 2.asm-
lbr2lim sin I r'+0 {
- ;-,0-'2 = 2 =L-. -txllm 3!n-.r_+o z c tim -___L*+0 [
2
1d1, li. *'o- *t'- tapm.,-+o x-a
Holll Mefum cpvirmelerdan istifada edok:
'* tirl iio"= lir, , tbg;tirf
x-+o x-a r-+4t.r - Arsrtrrsl.tr a
Burada r-a=y avezlemasi esrsaq, onda /JO olar. Bunlan axrrncrifadada nezera alsaq
- sina-sin(a+r) ..'d'-;)tt;ultl -----:---r-i--:- = - uE -----_:-7---\--:- =
),-+0 ysin(4+/rsrno /-+0 ysD(a+ /rsln 4
=-zrinoos{ a+!) r*'bi.r. ,1., =-To.y-o \ 2 ) 2 y"ro .y /-+0 sitr(a + /Fir a sin' a2
Hellt K:r;rm suntinda lazrmi qrupla5drma apardtqdan
rioonometrik cevirmelar ta6iq ernakla alanq:
,,- Le(, + z,) I rc(o * 1)l- [rg(, + ,) - re,]
=r+o X2
sonrq
4l
- ,r^ Exlt + Eb + u\cG + ,\- EFII+ ryG + ')' Eaf -!+0 r_
= !\%bs@ *x\z(a + t)-8(o+ x\'Eol=
= ri,n'F'q!"*d lsb + u) - tgal=
= fi^ ts tsb ! x\e2.x .b+ tg(a + zx). Eal=r--'0 x'
= tuntg- u,n 8!I 1;sr rg(a+x) [m [t+rg(a t2x\'ryol=,-+O r r-+0 r r+0 r__to
t ,\ 2rsa 2sina= ztga]l + tg' o l= ------e- =
-
.' cos_ 6l co6- 6I
limlF= li-th' ', I
=lr+0 .r r-0 -r lm cosr
r-+0
limitinden isifada ediLni$dir.
l4s Liniti rn*,,,*Lg#*=qHotlT Yerilan misahn hellinda mueyyen elementar gevirmelsr apataq'
Belo ki, evvelce kasrin suretina cos.x- i olava edok, glxaq' Sonra uylunolaraq cos 2: - i alavs edak. gxaq:
.. I - casr+cosr- cosrcos2rcos3xlllll
-
r+o I - cos.l- I - cos2rcos3r
=l + limoos.x.lim ----._-r+0 ,-+0 l-cosr,- l- cos2: +cos2.r-cos2rcos3l
=i+run.--.---t+o I - cos-x
I - cc2-t -. -.. l-cos3,=l+ lim-+ lm cOSZr lrm
-
-r+0 1-cosr t+0 rr0 I -cOSr
z.in2I= t+ tim
zstn r+lim 2 ='-o 2"in: { *-o 2.in'a1)
| ,l l.i't l
=l+firnlt"l +timl 2 l =l+4+9=14.r-'d rl r+d r Ii qrn- | I stn- I\ 2) \ 2)
Burada
1,15. Lirniti taprn rim (in-,/; + r - sir Ji).
trIalll Owalca sinuslann ferqini hasils gstirek, onda alanq:.^n-.tr Ji+r*Jiuln zsto
-
r--rd) 2 2
=2lim sin ----:-l'-------=, li- "orfi*f =0.r-+@ 2(r/.r.1+rixJ:-+"o 2
/ --1tt2147. Limiti taom linl ' ' '
I' ,+-l. 2r - l,,6lalli Misahn hallinda 2 - ci gdrkanrli limitdan istifade edsk. Buna
g0re mOtarizanin daxilina 1 elavs edak, grxaq:
T ?'-r '1" '
,,"lf,*:-')-r-; I,-"1\ 2x-t) |L]
( x+a r-a\,-; 2s61_srn -___- |
= 1i.11.. 2 2 | =,-,1 sin a I\,,
1-r
),2 -,1 ]lru I=s'* 2t-t =, x' r' =e* =O-
I
14& Limiti raDm ti.ftin'')i].. ,_4sinaj
Halll Mbtar'tzenn daxilina I elave edek, gxaq, onde alanq:lt
t-L-|,:ry-r']l'--' -,' 1.,* sinx-sina)':' -
'.r,1 \sina )) ,-"\ sina )
221$lla
I f 2*r'+o"io{-')= ri'l ir+ 2, 21'-"1 l. s'!n,
)
,,216 -rn crfa um
rn 2.---L..']- '* l:!=e,* .nrd t-a =e 2
=eoAo,
149. Limiti taprn J*("inl..*f;'
22
43
HaIIL x -+ a, cosl -r t olduEundan un l'" { = ' oldulunu nazsro alsaq'
,'lrlalartq:
16(1'11'=,.r--\r )l5O Limiti taPrn lim (sinr)E
2
ruaAL x=t+ y evazlemesini aparsa{,, v-+0'
,t,=,"(; -r)= *,,r,'o ='rll*,)= -'oolar. Bunlan verilan limitde nezara alaq:
lim (cosy)-'g = lim^[r + (cosv - t)]@ =r'-0 r4t)
@tt_l
=u-{h*(.^y-rlaif 'e' =_;bt. .- -. ^- lzsit2r -I
=,i--",H' = "8 4fir*" =,,8,-i P'*" =,0 = t.
lSLLimititaPrn j5x"*'#.Eali Misah ikinci g0rkamli limite getirarek hell edak:
,,,l."-*)' = *[t.l*.4-,I' =i-:l--- J,I "rr \ '!n ))
t - , t({.""i,-')J= r-llr.l,.**-t)1"-;"I,--[L \ 4n ))
]
-. ^ 2 u -^rrt. ,2- l_utr zrsE_
-/- P- 4d _ ;=e n"ra 2'ln =e '- =e t .
Burada sin-t- -f, oldulu nezsre ahnmrgdr'2.ln z^ln
152 Limiti taPm,,1i#.rtl
Hatli Kasrin suret ve moxrocinde m{teyyen element{ gevirmelar
ryarsaq, verilen limiti a$Edakr $ekilda yazmaq olar:
44
rn."(r * -1.)ri- \ "'1='-'ror"Ir*1,]
\ e")
ln
lnlim 3
,,
)
+.i4+
+
+
3.r
2xlim
)
2
e3
^2,
l+
l+{(
+
+
et'
"L
r53. Limiti taDn tim lncc(4
.. r-+o hcos6r
Halli Ovvelca kasrin suratinin limitini t4aq :
(_ t I cdar-ltim tncosc=tn tim{[r+(cosc_r)ffi; I =r+0 r-ro[' - "-
)
= 6rJ5(*'--D _
"r-Br*'? = _ rim zsia29..
Eyni qayda ile maxrxda mueyyen g"rir., *a,"rJo]o"rrnq, 'lim ln cosDr = tim - 2sin2 4.
x_+0
oldulunu alanq. Belalikla,
.- - -,, IE- lrm zsin- -*-+0 2
- lim 2sin2 4r-+o 2
(.c\2
"2rl r.l.
I
tim sin29 t lEl= ;llo"- ? = n^ \ n ) =o' =(!\'
lim 2sin2 ?{ :-+o , . *rz 6z \t ) 'r'+O Z t2rl"T I4lEl\z)
r-+0 2
154. Limiti taprn umd -'" 1aro1.x)o x- a.IIalIi Kssr 0zsrinde elernentar gevirrraler aparsaq, alanq:
Ir*'-o I -r\ o)x-ct
oOnda verilen limit afagdak iki limitin ferqi olar.
d -f _ oo a'-o -l _ ro-l .
x-a u-a
i^d - *o = oo rint {::} - o,-1 \m
,-+o U-d x)a X-A ,-+A
Her bir limiti ayrrhqda hesablayaq:
Ir* t-o I -l\ c./a:l
a
liro "-'- t
*+a I-A
buradq r-a=y avozlemesini aparaq. Onda l-+0 olari
ti. o'-' - I = ,^oY -t -l' ..t=t'',.a x-4 t'+o y bt=l+t
v=bg.(r+r) 'l-)o I =lirllo#r=ha'
Ikinci limite Bemulli dtlsturunu tatbiq edak, yoni
(r*'-"\ =t* o'''o =t+x-a\ a/ a
oldu!undan
f:lg
I
-. ( o' nb' +"t\illiml_-l,*(3)
{r*l-.91 -r\ a ) l+x-o-t -x-o
a
ahnar. Belelikle,
li^o'-to = oo loo - oa-1 'o=oa1n4-ito =,-+a x'a
= a" tr.a - 11= ad 11na -h e\ = aa 69 '
tss. Limiti tapn i*4+:E (h-ol'
-llalli Kasrin suratinda m0ayyen gwirms aparsaq, alanq:
,,^n,.4ff!=;g1" ,[#lBumda-
oldulundan
danq.
156 Limiti uPm
-- oh-l .
l,\- n =^o
,r^ o,-o( u' - r\'
= o' 1n' ,,,ro Ih]
farA lkinci gorkemti limitdan istiMa edak. Buna gora moterizanindaxilina I elave edek, gxaq:
d".';.")* =,r4,. a'+b'+c'-3
I
a'1 - b" -o' - bt\;.-td +b' )
x-+\xM *+0 +b'
.. I I or' .t br' -l or-r tr-t-l=i5ld+dl-;r-'r+----- .r- r -, J
r,'id:.{,1-.'".l3dttrI
=eBuada
ri- tt-l=hro, Iim o'-1=lr4
limr--r0 I r-)0r ,--r0
oldulundan, axrrnct berabarlikden
,j{r,,rrr.r") _ "}ro
* =rJoO".
alanq.
12 l
157. Limiti taorn , r[ o" * a" ];.. ,_\ a, +0, jHaUL M txbanin daxiline I elave edek, guaq:
,-l,*tBurada sadelik iig[n
M=a'+b', tr1=o'2 l-6'2 -o' -6'igarelemesini qebul edek. Onda alane:
.ut M1;M .. N
ri.llr*lL), | =":y*,M,-t\. u, )
lin - i tanaa:,)o Xtl
.. IY -. d2 + b'2l|trr
-= ItrD
c'-l -
-=ltrcx
o)
.-2li- ,' =-l =ba, ti. D' tl =ua,t-+0 xl r'-r0 l-
6.9'J=no, 6.-41=116,r--+0 f r-)u r
,. o'2- l.r=r. ,"--l tiro-,=o,t-+0 xz x-+0 r' x-tu
a)
., '2ri- D'
=-l.r= li. 6' --l li. ,=o
x_+O xz ,-r0 r! r-tu
(2) berabarliklerini (1) -de nezare alsaq, alanq:
- lrur+ ual=-lkrab = -bJfr .2'-' 2
Burada
oldufu nazara
alsaq, alanq:
.. 11,Y"t\E=,
ri. lLalrnmrpdr. Bu axrmct tfaizni e''+o'M ifadesinde n5ze'.a
"P-**="-rJa =F =#15t lim(l - r)bs, 2 taPm'
r+l'
IIeltL, - | = y ) x= 1 + / evezlamasini apuaq' Onda y + 0 olar'
bE'2=V=lZ=W' lnr x-l Y
oldulunu nazaro alsaq:
lim(t - r)los,2 = limC Y)192 = -h2 't-o' -' J
olar.
1s9. ,*frr,ttl-'rupr.t-+0 t' -lIlatJi Limit altmdakr ifadeni suratin qolmasmarwrsaq' alanq:
r*p;ffiD=lnsff;=iBurads
ilt
-. sltr-rlun-=1,t-+0 r
oldugu nezere alnmr$[.
16{L lim f/?-r, tff)ln-,-o ln(c/r3r) '
HaIIL l-aztmt ebmentar gevintralor aparsaq, alanq:
z"r2!)rim u" #ir-r0
Burada
^[,1,I)zl 2lsn23r 4l l, I
lijn --=2= ,t \ z ./-!x-+O sfiza r--+0 ( s1s\2 4
l.;loldutu nezaro ahnmr$r.
6!11_16l. lim , ,l-r . taDrn-
,-+o sctg\t + x)- orcrg[ - x)Holll Oget
cmtgx - arcty) = arcrt#dilsturundm istifado ebek
,,.ctg(t + t\ - oag(t - r) = -u fibaraberliyini alarq. Bmdan baqqa
uEr=r"[*2L]l-r \ l-xJkimi yazsaq, onda verilon limiti aqatrdakr kimi yazanq:
Iim e' . l=lne=t
.. sh2 xhm......-.........-:,-ro lo(c[3x)
=lim-r-+0 fI
tnl[t +
t
rr[r+("4:r-r)]-,ot-
1tzsrz3l 9'
fi, ---2hte'ao shz'
49
"(,*2ll 2'r *o(,) . -2
rn \ 1= r/ = tirn I _-,f "
= fim',- ^ =2.:Y";ts:t'=u f;+o(')'-o r-'
Qalqnultt:
A$E dah lini ?i tapm:
ffi.o y:[i:1, b) rgt*' "),r*'#::Cavab: a)1; b)?;
,ar. r,, (r* rX + 4Xr+3r)-t.
rJO
,*. * (r * r[ - (t.* s'),
,ur.,il t,-ii5x. -,.X, - ox,-,t
'uu.rg*q#d.,ur. ,* (r*!Ir'*rtB *r).
[,"Y *r[,us.$,#**'t6s. !^-,##.tlo.!1#,*ir7r. r^":.'*';o'*'.
x-+2 xa -E/ +16
172. lin 'l-}-t.x-+-l x) -2r -7
DlCavab: 6.
Cavab:10.
Cavab: 5-5.
."""0,(i)".-CrD
Cavab: z 2
cavab: -|.
Cavab: j.Cavab:1.
Cavab: 1.4
cavab: ].z.rl0
csvab: l;.J .
50
g^us,b, &)o,-,2
,ra. ,, r*r-(z+ !F+ z,.+r G_'f
curub' & * l)2
l7g. lnn vr+{r+{r.r-+J& Jr + I
'*.,I.Lf#q.181. timJ+=2r-3.,+. .Jx - 2
rs2. ri, J--3.:+-t 2+Vr
tt3. fi- ffi,-2G.t-+3 r' -9
l'I. l:,^?tr:6'2,--z l+8
rss.,riEr-f=.
Cavab: f24
(a - natural sdeddir).
Cavab:I.2
Cavab: l.
Cavab:4.J2
Cavab: 1.3
Cavab: - z.
Cavab: !.4
Cavtb:25
,rn., lr'1-z'*t.,-l -y{ - 2r+ I
175.r (t - o')- *':t(,- o)(r-oY
(z -natuEl odaddir)
177. rim 1lf,+sl' *(, *4\' *...*[' * ('-'E)'l
,--zl\ n) \ n) \ z ))
,rr. ,,n[13+23t"'+r'3,-r.o( nJ
Cavab: - It6
Cavab: -!-.1't4
It6. tim"'q:2r-s.'-*
5''lx -2
187. lim.'[-zr-r'-(rn,
Cavab: - 2.
5t
Cavab: x2 +*+43
r97.,ri. (3r[3 +3,'? =tl=).
I200. lim p(.x+o1).(r+a,)-r!.
Cavab: 2'
Cavab: I .4
Cavab: a.27
Cavab: 1.2
Cavab: 1.2
cav*: j G+a).
Cavab: !.2
ICavab: - :.
Cavab: l.
Cavab: ?.3
Cavab: 2.
cavab:1.3
Cavab:-].
Qay6glL(a, + a, + ...+ a,)n
(z - natral adeddir).
,ar.r-ffij.1t9.
1m.
19r.
t92.
.. l!fi-?tn-Ilm
- -------:--.
.r-+O x+21x4
- .Ji+r-./i;ltm::-_-.,-+o {/l + .r - i/l - x( 1 2)liml
---------
I,-+t\.l-'/r l-Vr./m [.[x+ffii{-r].
ll, (ilt;;a-lF -7;
m.ri-.-(.f*fffi-J,).te4.,ly-,(.tii - rJT*' *,).
,*{r95.
195.
rl- 2 21
198. tim .rilr.r+r1i -1r-ry: l.,--'co L l
3
199, lim -il\lr?i-zJ, *t + Ji).
2n2. ,^(Ji? +'\' +b[t+] 'x\' (n-natural adeddir)..r-+0
Caab:.2n
52
Fffi)
Llnti ari tapnl
203. li- ths'.r+0 r
2(N. lim sior.
205. h" l-T'.r-+O xz
206. tirnt4.
2M,limrctg3x.
2og, lim 'g -.sio ' .r+0 sinr r209. l. sin 5r - sin3r
.r+0 sin r210. li,n
*t'-:*3'.r-r0 x'
zl. n+tsu,e(i-).4
2tL t[n(t-xys+
213. B*aberliyin dogulutunu isbd edin:a) Iim sinr = sin4 b) lim cos;= cos4
v) !:L,sx = tsa (^ * !| o " = o,*, *r,...).
Llmilladtapn214. lim sinr- sh a.
,-+a x-d
215, Iim cosr-cca.x-+a x -a
216. liaB-t4a.x-)a x-a
217. lim"tg-"tgo.x-+o x-a
2lt. [- s€ct-te"4.,-+at | - a
Cavab:5.
Cavab: 0.
Cavab:12
Cavab: I .
Cavab:1!
Cavab:
Cavab: 2.
Cavab:4.
Cavab: 12
^ .2LAVAD: -
Cavab: cos a,
Cavab: - sina
Cavab:xc2 a(o *L1zt* rlt =03112,...)
Cavab:-fr (a + tr,r -tam adsddir).
12
53
cavsb.i'r+ 6*l(p+r\woa 2
,r. ,,*oklr=yry)ll9',,ro. * cos(a + zx)- z o9s(' * r)+
"ma.x--r0 ,'
221. ti^ "&(o *2')-u,e\ol,43:E
1t-,--t0
Cavab:$ (a+ tz,t -tam ededdir)'
226. lirgrJ0
227. li^ l-"tg' ^
.
,)!2- ctgt'ctg- x
22tw^@P.,2
229. lim _---=:rro./l +.rslnr - {cos,
34
,rr,,. sin(a + rhin(a + zx)- sin2 a
,r3..'t]#Hi*.6
- ( zr\tstnl r- 1l
224. hn* .r I - Zcosr3
225.6ffi.,-icotl +
e.]
teb+x)tg1ta-t)-ts2a.2r
230. li,,[T-r'Ja; .:-.{) Stn- r
,rr. ,*{:;7t+0 I - OOS.X
232. tir]:@.,-,01- cos(Jx)
r -tam edoddir).
Cavab: - sina
Cavab: - cosa.
Cavab:lsin}a .
2
Cavab:-3.
Cavab:l../3
Cavab: - 2l
@32aCavab: - ----7- .
o(E (I
Cavab: 1.4
Cavab: ].cavab: 1.
3
Cavab: - I.l2Carab:Jl..
Cavab:0.
,rr. 1*t=aff@t-"[
zu.,r('*')r-.'-+d\2 + x )
r_G
zss. '*(-t:l'l
- .,-r\2 + x,/r- Ji
23d. rim lrr:).; .r++42+ x )
,1
,rr.,' {,r'l -'+r)r; .
,-+{ 2rr +.r + lJ
23s. [-frin' 2- ) .,--\ 3n+l )2Je. ry !g{l+x11's2'.
,no. ,*['1 -')'' .,--(rr + I J2
/.\l
241- ti.n I ri +l I
,-*lxz - 2 )I
,n - ,. ( i_+zx-r); .,-\L\, -3x -2 )
213. lin!t-2x .J-+0
244. h-r:+4)' .,-.-\r - a )/ . \r
245. rim i','*l I (a1 >0,a2>o)x)+c^ s)x +D) )
\_!LCavah: qca. olduqda 0, q >a, olduqda @, at--az oHuq& e "t .
zlo. ti.^(t+r'f" . cavab: e.
247. limf + sin zaXso . Cavab: e I '
Cavab:3.
Cavab: 1.2
Covab:
Cavab:1.
Cavab:0.
Cavab:0.
Cavab: 0.
Cavab: t.
Cavab: e3.
Cavab: t.
Cavab: e-2 ,
Calttb:e2o.
55
I
2.4s.li-r I +{r l-'r+0\l + sin}/
t
250. limr I +'ls )d],-- .,-+o\ I + sin.r_/
I
2s1. fimr *sx ),
r*o[c62r,/2f,2. t,r.QerYL.
4
*r.$,|,r(i-r]*2s4. rimif,;f.
r+0
255, ti.r'+')a-+o\z-1,/
255. [, h(l * ').t-)0 ,257.
,lim .rffi+l)-hr]
25t. hn lnr-ln4 (4>o) .
t-+a X-A259.
,tirn [sin rn(r + l) - sin hr] .
260. tt, "l'l-'*'l .x-r+o fu(;ru a 5a 1)
2dr. n,,f h rm *'1'l,-+-(-l+l00rrJ
252. tim tf.€.El'-tr'6 lnu + {/, + trr,
,r. ,* le(x+fi)+le0-n)-2lg.r lr r r;.
h-+O ht
Cavab:1.
Cavab: .6.
3
Cavtb:ei.
Cavab: e-l,
Cattab:e-2.
covab:1.
Cavab: e'*1.
Cavab: l.Cavab: i
Cavab: f .
oCavab:0,
Cavab: - 2-
Cavab:
Cavab: f5
3
,
25/. hnr+0
cavat: -\ff
cavab:2f
, (z )''\o** )
trr,56
,*. ,,.,[t-*ffi] .,-.t ,+^lt_r2 )
,.o. ,, roh*.I--,'{J'-o hF+Jl-'2,
zst. r44-l 6,o1.
zOe. rg(,*,'i.I
2d9. I .f 1 *, 2' ',]7
,-(r +x.3, /
,ro.'',,Il +tio'*'-Jo"',,-o(l + sbrcosft/
zzr. ,r ''[-i) ..t -+ t 5i1[zar',f
,?r.I*;frt3n .
274. li- "- -"8 ,,+osi!6 -sitrA
2is. 'tim'".- a"; (r, o).xlo xP - aP
., .b276.t\; (o,o).
,rr. ,nG+ of '"(r-+ bY.'b
,-- (a+ a+ bf,..,.278. ttri-44l;-i (,'o).27e. ttnnz(li-"y;) G,0.
zeo. ri-f'-r +Vl'i (a>o,D>o)'+6\
a )
2sr. ri.|,6i{E) (a>0,6>o),+o\ 2 )
Cavab:0.
Cavab: z.
Cavab: lna.
Cavab:e2,
Cavab: ? .
3
cgvab'. "l'*' .
^.duavab: -ICavab: - 2.
Cavab: e2 .
Covab: l.
Cavab: a f-tp
Cavab: aD lna.
Cavab:rt*6).
Cavab: lnr.
Cavab: ln.r.
Cavab: "JE.
Cavab: Jat
2r.. r,.(,*tat+l + bt+l +ct+l
I
)'(a> 0,6 > 0,c > 0)
a+b+c
cw*:Q"abtfi*;
2tr.. i^{ -b': (a>o,D>o),-01", _a,I
2sa. \^.+ G,q
,*,'gi:t#,*a. ,' r"1r + l'l
,.+*- ln(l+ 2'J
ztz.,rr, r,(r-r),(r- 1)
28S.,lim {=o (a>Ln >0) isbat edin'
2g9.,li, !99t'=q @>!e >o) isbat edin.
c"u*' (n9f'\ b)
Cavab'. ao' lta
Cavab: 0
cavab: !1br2
Cavab: lnE
Cavab: I
+t)+:rhr] Cavab: o
) Cavab: lna2
^lLAVaO: -8
I,
I,
'+tlr-l I
58
,*.ls*E +k1 Cavab: - 2
Cavab: e2
2
Cwab: el
lyr.
29t.
299.
,nf,, * -,30:i, lim
3m. [Inq5r+0 r3ol. lim
c'x;- It-+0 a'
302. fin|4r-+0 ,
(a>o,P>o\ C""*,ffi
Cavab: cha
Cavab: .rha
Cavab: - 1
Cavab: h2
Cavab: t
Cavab: e"
Cavab:
Cavab:
Cavab:
cht
Cavab: I
Cavab: I2
Cavab: I
ICavab: 2shj
1
E_,
3
2
304. h, "ir - "'"',+o x-a,n, *cls- clP
,ao l-a
306. tirn h"'h
r-+o ln c,c r307. lim (r - h ctc)
ri!2r -air,308. lim ' -'
rJo tl,a
310. rin rcsinll
ilr. ,ri, *""."("f,t*I -r)
3tz. traa,asffi
, rllI E\lch-lJog. linl n
I
"+.1 n Ilcos -l\ n)
59
313, lim orctS'+il+r'
3r4. ))oq!&+-arc!s-s15. rim [r-.rg-rr-:t -
re
315. 1irnf,Z-*nr:=)r-+@ \4 r+t)
317.,rm,[;-"*'t#J
"l+"*))
. ,*,"L.I;7)us. ,*lr*(-t)'I ' )
,-.1 n JI
stL rm $"-,'320. lim.rlnr
Cavab: o
Cavab: o
Cavab: I
60
rzr. ,ti. ({'*r-fnz. ,*$*.-.)szs. .ri- (.[*,*7 -.tl-;;7szr. ,u- (.[*I*7 -.t1-;;7325. a) ,linoacrefi
b),rigaraej
SZe.4 ,rit ^-!
l+ e'
b),9*+l+ er
327.Q,rim !!@
tl ri'" Etl4x
Cavab: Z4
ICavab: -l+f
cavab:i|-t'+l,
Cavab: i'2
Cavab: 1
Cavab: ei
Cavab: o
Cavab: o
Cavab: + o
Cavab: I2
Cavab: -1
Cavab: 1
Csvab: I2
Cavab: - I2
Caveb: 1
32t. limr-Efr-+o Y x
rzr. ,rm
*.(".6+r)
330. ,lim
si"'?E"t';)
Cavab:0
Cavsb: o
Cavab: I
$ 4. TUNXSYANIN KOSLM0ZLiYi
r0. TUbq ki /6) tunksiyasr ,r' aflprlLgmda byb ohmmq
funls iyadu. .r0 ndqbsibuaalr$ daxiHir, (a ex).Bu antqdan r0{ yE an \,x21,.a11r,. noqttbr adIcrllrEml sefet"
Onds bu noqbbde frnh iaam ahgr qiymedor adrilhttfG.lf!r\.../G,)... ohr. Oger n0qbbr adrrll{r ro{ vElldqda
funks iyahr adrc tlr /(ro)-a yrlrhsa, onda /(r)a r0 nOqbsidekesitnaz fi'aks iya deyilir va
n rG)=rh) (1)r-rro
kimi yazdr.Bu tarifi "a-6 dilindc" sdyhyok lxtiyari mlbbet a >0 qaSr eb
mllsbat d > o edadi vaBa ki, x aalsma daxil ohn ve F-41<,be abe s zliyini odayen b0fllo r - br 090n
[G)-/Go)'"benbes izliyi 0daosin.
/(r) frnh iyas nrn ro nbqtss ftda kasitnezliyinin
flgtbcll 6rifi yazmaq ohr.(1) brifloki /(a)-i sol bo6 keginib, hsmin benbedivi
nm [f(r)-f(xo)=o (J),-tO-+0
kimi yazaq. r-x6 Bqi atqnment attmL /6)-/G0) Orqi le flsb iya
arhmrdrr. Bunhn uygun ohraq Ar vo Ay kimi yani
Lx= x- xo,Ly = fG)- f6r) ban esek, p) beabediyinilimAv=0r-r0 '
kimi yazarq.Ba$F s0zb arqument artrmt slfu yaxmhrgdqda fink iya artmt da
r,6v 6664 fimks iya kesiknszdir.
51
(2)
r-ci 6rifi esssnda
(4)
Ogerbu ts riflar tHanmisa onda /(r) firnla iasr ro ndqtes in& k.s ilir.
/(r) frnks iyas nrr ro n6qbc inde saE ve sol limitbd vasa, bir'birin
bember deyibe, yani
,!,,,.fG)*,S_fG)onda bu kss ih birinc i nOv kosihn adbnr.
Sa! ve sol limitbr vasa va birSirine baqberdisa, onda bu keeilrm
andan qtlhnh bilan kas ib adhnr.
Ogar R n0q!s hde firnh iyann birtoofli limitbrindan biri yoxdusa
rc ya somuhrlp beqberdise 6a1fu 6rnkr iyaon bu kasikrczliyi kinci nOv
kasitrp adhnr.
20. bciyari 6>0 qEEr eb d>0 ededi vasa ki, ixtiyari iki ./,ra e xn@bri Q{h V - 4.a berabes 2tiyi odeoildikde
V€\-tQ'[." (5)
be abe s izliyi Odanilirse, onda ,r(r) frnksiyas oa x alo[Emd8 mllnt zem
ka iLnezdir &yilir.
331. t$)=Zi -Sr+,1 frntc iyas rnm (- "o,ro) aatlmm ffiari .x6
n0qtrs inde kx ihaz oEu$rnu bbet cdin.
E E Limitin bdfinden btiqda etal onda yazanq:
[m .rG)= ri. (zf -sr+l)= z163 -sr6 +4.x-rro ,-+.t0 '
/Go)=zto3-sro*4 x6 n0qtasinde /ft) fimls iyas nn qiymetidir. Bu
onu g0s5rirki /G) fuls iyasr t n6qtssinda ks ihzdir.
ii2."a-6 dilintla" a) u+b,b) 12, v)sinx frnksiyahnnrn tasihez
oHulunu b bat edin.
62
ECll a) a+b ksilnaz ohu$unu bbatedakva>0 090neb Vro var
ki,
@ + O - cs - tlt= d{-tol.", lr- tl. I = alr,ro\
Udaair.
b) htiyari a>0 gofirnk Onda
l" -,r'l = lG -,.I * z,ok -,0 )i = l, -,of + {a!r - x6l < e
busdan
;r-a|. fijt*" -1,.1=0G,,.)
v) linr-sinxsl=|'-i"*t&]= z.f rel=t,-,0l<e, 6=e .
JiJ, /G)=+J r'ro ve 7(o)=t. Funls iyanm kesilrrazliyini bdqiql{
edin..IIaIlt Birtaefli limitbri yoxhyaq:
-. sinr -. sinr -l[!1 --;-_; = llltr-=Itr+{4 lrl r-+0 r.. sin.r -. sinxtlltr --;--; = Itm
-=_t'r-+-o 14 :.+-0 -r
Beblikb, r = 0 nOqae inda funh iys 1<s ilir. Bu birinc i nOv ks itnedir.
,Jr. /(r)= sitrl, r*0 firnls iyas mm ksiknezliyini odqhedin.x
Heltt *,= t ,r;=: I (n = t,2,..) obui. a-+.o olduqda r, =0 \,atm ,!h+n \
2'rl =0 ohr. Odur ki, /Q)-+0, f(x:,)-+I. Bu onu
yoxdur. Bu finls iyanm ikinci nOv ksitn$irir.335. l(x)=z+ --J-- firnks iyas nm kas ihrazliyini odqil e<tin.
l+28
gGtarir ki, lim sinl,-+0 r
63
Edlllfunls iyanm
hpaq:
r=l nogbs i kos ilme noqbsidir' Beb ki, r=1 noqbsinde
kasr hbiesi srfm gevrilir. Owebe sol limiti ,tim-o/G)-i
.r+,ra> J{'. ;h} *-,['. i)'indi saE limiti hesabhYaq:-()r')
rim -r(x)= ri. lz*--f-l= ri. lz*-J--l=r-+l+o a'+q ,1 -. I "-d - l' I
I l+2l-(l+a)J \ l+2 d./
rltt= ',-
L*--.] - l=r."-ol ,.+
I
\ ,;)Beblikb, sol linitin /0-0)=2-ya sa[ limitin /(1+o)=3-ya benber
oldulunu aHq. Bu onu g6sbdr k! r=l n0qtasi funksiyanm birinc i ndv
k* iftne nOqbs llir.
1_ I336, f("\= Lt'+ funbiyasmm kesilmszlivini mflevyenlidirin ve
r-1 x
)(8nktsrini g66dn.
Halli x=0, ,=-1, a=l n6qbbrinds fimksiya byin olunmayrb' Yeni
bu nthbbr kasilms noqtabridir. hdi her bir noqbde saf, vs sol limitbd
tapaq. Owebo fim.lsiyamn gaklini deyf sk'
fG)=*
r=0 noqtos inds fimksiyam bdqiq edak:
tim /(x) = 1i,' t]=-r'
r-0+ x -+o+.tr + ttim /(x)= ri.4=-l
;-+0- .x-ro-r + r
, = 0 ndqos i aradan qaldmla bibn kgs ilne n6qbs ktir.
, = I noqbs ini yoxlayaq:
Iim /(r)= tiltr r-l = [m l+a-l =0rJt+0- r-+l+0r + I a-+Ol+a+l
[m /(x)= [, '-l=liml-a-l=-li- a =or+t-o- " ,-r-0r+l a-+ol-atl d+02_d -
r = 1 nOqes i de amdan qaldrnla bibn tas ilms ndqbs irlir.
Nahayat, .t=-1 nOqbsini yoxhyaq:
lim /(x)= lir" += 66 -!+a-t- t^'-2 = rirr6-?l=---,--r-l+0 x-+-1+0.x + I d)1-l+A+l d+0 A o-0\ a)Bu n0qoda sat limit -co oldulunda4 hemin n6qta ikinci nOv kasilrre
ndqbs i adhnr.
337. f(x)= cos2 L frnlsiyasm.rn k*itnas hi ve k s ilrnenh novlinux
byin edin.
AeW x,=), b=t,2, ) obrm. n-+- olduqda r,=s olur. Odur ki,
z-ro olduqda cos2a-+l ohr. r; ==-L b=L2,..\ obrm. z-+coxn \$*t')
oHuqda ri -+ 0 olur. Onda n -+ o oHuq& *r2 ] -+ o oHu[unu almq.xn
Beblikb lim "o.2
] yoxdur. B u onu gG torir ki, .r = 0 n0qtos i ftnts iya ll9lh,+0 X'
ikinci n6v kesilrm n6qbs ilir.
,J& /G)=,I-l*#(.r > 0) frnls iyanm kes ihrzliyini t dqiq edin.
55
Hatu f(r)=
.r = r birinc i nOv kes ikne n6qbs illr.
ru. fu)= ly-i!,.{ firnksivanm kasihnazlivini odqiq edin'
l. ^HiuL x>|orduqda 7(.r)= trm 'ii;{ =L; -{:==-"
r=o orduqda t@= 1y1)ff=o;
x<o orduqda f@=m\ff =,.
Bu onu g0sbrirki, firnls iya r-in btflln qiymatbrindo kasilrnezdir'
3a0, f(x)= aragliigtln 'x=o nOqbsiharcrndv kas ihna ndqtes idir'
HalllBur:r- gia sa$ vs sol limitbri tapaq:
tim a'ctgL=L, 1$o)=i;x-t+o 'x 2 Z
'Y\*"'tl=-i' ^-q=-+'Saf, va sol limitbr var. Ancaq /(+ o)
'r 7(- 0) benber deyil' Odur ki' r = 0
ndqtas i birinc i nOv kes me n0qtas llir'
311. lG)=Y tunkiyasr 09iln x=0 noqbs i hals r nov kes ilme
nOqbs idif
*#[0 (.x l1
r>1
66
flelL x=o n6qlos inde funt<s iya kesilir. Ancaq lim!9I=l var.Buonu
g&brir ki, funksiyanm sol vi sag limiti var. Yeni y(-o)=7(+o)=r.
Demoli r=0 n0qtos i funks iyanur andan qaldrnla bibn kesikns ndqbs llir.
312. f(i=! tuds iyas nm (o,r) anhlrnda kes ilrnez ouulunu ve
m0ntazam kes ilnaz olrnad{rnr gds brin.1
HallT : &nhiyasr ebm€ntat frnks iyadrr ve byin obbstmdax
km ilrnazd ir. Ancaq (o,t) aahf,mda mttntazam kasikroz deyil.
"'=!' "'=-+n n+28
g0tilok. Onda a -+ o olduqda
Iakin
14.'!4.")=r,- n-24=2€ > e
Odurki (o,t) arahlmda mtntazam kas ilrnez &yildil3{3. I - ixtiyari m0sbet odad olduqda 7(x)=.r'z funls iyas nm a1(-t,t)
anlltmda mtinbzcm kas ilmez, A) (- co,+.c) arahf,mda mtinozem kasihraz
ohnad{mr gCb terin.
HalE a) ve >0 rigiin eb ix'- r] < |=d(a) vark!
IJV\- f(rl=F', - ',21=l', * ,1.V' - ,l<?y'! - x,l< e
beabes izliyi 0danilir. Yeni (- t,l) aralgmda ./(r)= r' tunksiyasr
milntazem kas ilmazdir.
67
b)/=z+l v3 5'=a g0$rgk n + coolduqdan
.tlr'-r'l=--r0'
k6.')- r('"")=l'* *'2- * ")i "o"24 ='* * ";i;' 2
6t
VVI - flrl=ln'1 + z * ) -,'i= z * ), z, "
.
teli firnl'siys (-.o,r-) anh[mda mtlntazam kesitnsz deyil
311. IQ\=;j tunks iyas nn (-t<r<t) anhlmda mtrnbzom
bs ihez olduSunu gdsbrin.
Eattl (-Ct) aal$Lnda /(.r) funls lns rks itnazdir' Konbr Eoremina
gOn onda hemin aralqda frnls iya mifnbzam kas iftnszdir'
JrS. /G)=Y funkiyasrnm (o<rcz) aohprnda miinbzem
kas ihazliyini YoxhYm.
Eelli o<t<n aallmda l@)=1'f(')=o' yani tunlsiya [0I
pa4as n& kesihazdir. Onda Kontor Eoemine 96o hamin aahqda
m0nbzam kas mezdir.
316 f}=e'cos! fi,,}" V" " (o<r<t) inervalmda miinbzem
los ihnztiyini todqiq edin.
EeaL x, =*,r,' = i-r^ gdfliok. z-roolduqda onda
li -*"1=;fi;n-,ohr.
Yeni funtsiya veribn aalqda m0nbzsm kes mez deyildir.
i17. tG)= Ji funls iyasmm (t < r < r-) arahlnrla m{lntazem
kes ilrnazliyini tedqiq edin.
Ha i lr' - ,'1. a =s olduodall
14 )-4') =t.r- 14= ffi =F' - rt..,
Ysni ve ribn a mlrq da lG)= J; frnl$ Vas I mtlotozrm kas ihzdir-
Qalqmalan
Agafidah funksiyalann kasilmazliyini tadqiq edin:
34t /G)= li . Cavab: kasilmzdir.
34g. IG)=..: | -.; (r+-l olduqla ve 1(- r)-ixtiya ridir).
(l + r)'
Cavab: x=-l kes ikne ndqos idir.
I
350. /(x)=e-7 (;r*0 ve .r(0)=0). Cavab: tes ihazdir.
i351. /(r)= 'i (r*l olduqdave 7(t) ixtiyaridir).
I + ,lllCavab: .r = I kesihs noqtss ilir.
352. 1Q)= 7Yrx2 (x+o va f(o)=o).
Cavab: a=0 noqtas inda kes ihnezdir ve a*onoqbsinda kas ibndir.
3s3. /G)=J;-t.frt.Cavab: r=12 (i=1,2,...)noqbsinde ks ibndir.
59
AEa{utah luaksiyaltnn has me noqblerint byin cdh va ndvanl
uag&rut"
354. v=--j-=. Cavab: r=-l somtz kasikne nOqtas idir- (l + r)'
355. Y=-r-'su! r
Cavab: x=o aE&n qaHmla bibn kes ilme noqEs i' r=tz(t=i1,i2,...)sotlsuz kas ikne noqtrs ilir'
355. v=sm(gin4).r
Carrab: ,=i (t=t1.i2,..') birinci nOv kas ikne n0qtas i' r=0
ikincinOv kss itm n0qosidir'
-l337, y="lxtag:-.
Cavab: r = 0 andan qqldrnla bibn kas ilme n6qbs llir'
ASafuilah filliks$alarta hasthozl$ial aruSdnn'
35E' Y= lim ji---:;'
a+q n' + n
Caesb: /=sgnr; .r=0 birinci ndv kas itne odqbsidir'
3sg. y=ttn|6+F.
Cavab: l.rl<t olduqda v=1; lrl>l olduqtla 'v="' Kosiftnez
funls iadu.36O. y = llalroctgQrtg)|.
Canab: tz<r<tr+l oldn.{ld^ t=!'; ktr+L<x<kr+n
olduqtl_ y=-Lx; r=m*l (r=oJLte"') olduqda v=q' 5=la
birinci nov kas iho noqt s ilir.
361. v=,lim lo(La€"').
70
Carrab: r<0 olduqda y=6; r>0 olduqda /=!. Kcsilmazfilnl<siyadr.
l2x- o<r(t-X2. I@ = 1^
' _ ^ Cav-ab: KesilmaT firn[5;16 66y1.
12-x, l<x32.36j. a_nm hans' qiyrnatinda
(,./t-)=l' ' r<o'
[a +r, r)0.fiuksiyasr kssilmoz olar?
Cavab: a = l.M. Afafrdak funksiyalann kasilmadiyini araqdrm ve kasilma
ndqblsrinin nfidnii toyin edin:
") /t,l={"' osr< l' . D rt'l={t Ftt,[2-x, t<r<2. tt, l!rl.
d l.,=["8'o,.rramoldrda,. d) /rrl={rho, r rasicnal olirq&,
[0, r tao dd]qCa. [0, r irrasionl oldrda.Cavab: a) Kcsilmez funlsifdr4 b) x=-l bbioci n<iv kesilme
noqtasidir. q) r=t (t=0,rt,r2,.,.) sonsuz kssilme noqtasidir. d)-r *t (t =0,i1r2,...) ikinci etv k silm. noqtosidir.
Agfuh fuakiyalann verilniS oblasthr& rfi ntrzom basilnaliyinlarasdnn.
355. ^r)=Ur
(0<rcl). Cavab: mtmpzsm kesitnaz deyil.#6, l1x1= 6c1g7 (--o<r<+@). C.anab: mtntazem kesilmazdir.367. 1x1= rr,raa (O<I<+@). Cavab: miir azam kasilmoz deyil.368. Vsrihrt Ve>0 adeda qars 3d>d(6) edadi taprn ki, verilmig
arahfla f G) firnksiyasr tflln mtfi.zem kesilmazlik prti Odenilsin:
r) /(r)=5r-3 (-a<r<+oo); d) 7111= 2siar-o*.r (--.o<r<ro);
v) /(r) =l (oJ<rsl);x
e) /(r) = rsiol (r+0 vo /(o)=o), (o<r<r).rCar"eb:a) a=!; I d=opt6; 0 a=i; e) a=r-ti,$t
71
trFOSiLBIRDoYi$ANLI FT.INKSIYANIN DITERENSIAL EESABI
s 1. A$KAR nINKSIYANIN ToROMOSI
t0, Fsrz e&k ki, y = 7ft) firntsiyasr (a,a) arahlmda tayin olunmugdur.
ro e (a,6) r@asinda r arqumentinin artmr Ar =, - xo kimi igars edak.
Onda A/(&)= /G + a')- l(,0) frrqr /(,) funksiyasrnm 16 ndqtasindeki
afirxdr.Ogar
u- ryle) = r(r,) (ar * o)
r-116 &varsa, bu limit 7(r) fiuksiyasmrn x6 ndqtesinde tOremosi adlanr.
Ogor 7(r) ftrksiyasr ro ndqtesinda birinoi n6v kesilmays malikdirsa,
onda
/-'(&)= fro/(ro + ar):'r(& - o)'
r,'(*")=^,t - lhlA):l(+:!)
ifadobri uy[un olaraq 16 niiqtosinde (r) funksiyasum sol ve saE
t6ranroleri adlanr.Qeyd edak ki, Ax art[ru ixtilari qaydada sfra yocmfur.zo. Tannonin asas toplau $ldalul Ogsr c sabit aded u(r) l'(r) va
r(r) funlaiyalan diferensiallanan furksiyalardrsa a.Sa&dak ar doSudur:
1) c'=o; 4) @"1 =u'v+v'u;
ztQul=a,,: ,(i)=#r,,.)3l (u+v-wl =u' +l -w'; e1(r'l =-"-rr'(r- sabit edaddir);
4 Oger y=IQ) vo u=di fiEksiyalan diferensiallanan
funksiyalardrsay',=vL u',
dolndur.f . Asas dlfiiilstr Q"l =**' r D( (*nwl =-=!'' l+r'IL (sinx)=cosr; X. (r') =r'lo, (.ro) ;
72
III. ("ourJ=-rinr; XI (bs"rl=+ 610;
IV. (rF) =-+; XI. Gr,,l=a-;
V. Grs,i=-'f' XItr (cAr)=slrr;
Vt. (nqrirr=+=- )trV. (rn i=-L;' Jl - r' chzx'
VIL (",""o'r=-+; XV. ("wl =-*;,h- x' sh'x
vn. @"w)=-L=.
369' r arqumenti G{an 1000-a qa&r deyigdikla y = lgr funksiyas'"mAy arUmrna irylun r arqumentinin Ar arfirum taprn.
Halli t x = t@0 -l = 999, A,, = btmo - bl = 3.
370, Oger r arqnrnenli o.o1- d.n 0.001 - a qadar dayildifds, y=jfunksi)asrnrn A / arumlna uyEun r arqunentinir A r artmrDr tap,t,l.
Haltl L x =0.001-0.01= -0.009. a r=--!.- . --! - =sC.rOt.(o.oolF (o oD'371. x byisani Ararturr alrr. A5agdah funksiyalann Ay rut$uDr
tapm:
al y=a+b', b) y=*2 +br+";v) y=o' .
Eolll z'1 Ly = a(r+ M)+ b - ax -b = @+ olx - @ = a|rl'b) Ay=a(r+Ar )2 +b(x+Ax)+c-o.2 -bt-r=ob' + 2rtr+ (&)2)+
+ br +bLr +c - c2 -br-c = 2x\xa + o(Lr)2 +bLx== Q..t + t)L, x + a1L,r)2 .
v) Ly = s'+M -r'=at(ab -l).
3 7 2 f (x) = (r - t[t - z)'? (x - tl verildikdo t' (t\ l' (2\ f A\ - $ tz47m.
lTaIIi Tdremanin briftt g6re
rr,l= L*r@tr1*&d = lj$ ** -,#. - r,'
= _,,
7r
J.(2)= Iho (2 +
^, - lx2 + q- 2)'?(2 + ar - 3)3
= g*!Je$tq:u. = o
r<rl= i-"g4=U@3 + A' - 3)r - lim (2 + &xl l&)2(&)r - o'
ASfur gakiua verilmit funlab)sn n Mranesi :
t:-ins.,=1ft. yL=l
.Ealtl Owalcc verilen fimksiyan-r olvensli Sakilde ,aaq:t
ft *,3')l'=[-J
, r (r.,'f? f r *,3) - r l.r*,3 1-i l,'(r -"). *(r.,'')-Y=r[,-7J l-7] -3l.,-F ) i-tY
I
lr*Jlr r zx2 - u2 ,ft*i=l.,JJ irot 6;7=-=r-l-13
s74. y=zBI , y',=1 .
BarL y,=zcf, roz.(-:) =fi", # [ii = -:; + r.ixX
t-----=:-
375, r,=il+3Jl+1h+t1 , y',=1 .
Eolti Owelca frnksiyam elverigli gekilde yazaq:
lndi toremeni tapaq:
22r ft * I \-i 6x2 (r * /)-i zx2=;l.,-FJ IJ]=l*FJ (,-lf
71
t-i...-zr/r+'./r*t/r+ra
I
zr/r +3'/r + {/t + r'
:oJr *'/r*ffi .'J[. t,ii7] t[.,T
376 y=y1f 1f' y'r=l (rro).talli n = i ve y2 = xt' iNra erlgrek, evvalcs loqrifoalayaq sonra
tdremmi@q:lnn=fo;r=1!a1,
la = Lrr + I + r,r' =4(n.r +t)= rr(tnr + l),yt
lnY2 = x* 1o' '
fr =6'l ^,
* -''1 =,'(lo, + t)to, +,'-r,
rz = rz[.'(t, * r)U, * r*, ]= r,' [116r + r)r,x + x'-r].
Verilon fir*silaon tthsmesi y' - (r)' + 6'I . (r* ) oEugundan alanq.
y' =1 + rl(hr + l)+ r" ["'(u" * r)n, * r'{].JZ. s r,=Gin-rf6'+(coox)m' yr' ='tEatlL (sinxfft =y,, (oosrfb'=y2
ile igars e&k. Her biriain tri,ranresini a),nca tapaq.
,1 = ("inr)f* ,i.h&sini lryriftuhpqr
,
hlr = cqrlnsinx, lL = -siaxhsinr +*tr. ffi ,
75
Fl'l
]'['.'''."ij=
= 1(,.,'E (.r)=
{,.r,.u.,
i['.t-'FI
or/r *1/r * {r *,a
*' = r,[- *,',.,. *) = otn*'[* -'t""";y, = (ca"f ,
ifrdssini loqariftnalapq vo t6r,meni tapsq:
ln/z = sinrlncoo x '4 =*"rh.*t -'io' '
th',y2 "o0,
v2 = v2l*,'-o*'- *J = t""' o"'["* "*" - ll")'Verilcn finlcsifa.orn t&amesini yaaq:
Y=[,-,1*'f *["*')*'l =n'*Yi =
= 1"'a*,[# - si'rusi.r) -(o-,]-'[* - *,,t"-")
37& Verilan firntsiyanrn t6nmasini taprn' Funksiyanrn ve tdremenin
orafikini qurun:' "1y
= 14; u)v=#; 'lY=t'14Ealll a)l,l="en' ona goro Y'=asnr ('*o)'
b) Y = t' "gt*, Y' --2xsgtx=2ff,
J79. Comler [gun d[surr glxam:
PnG)=I+Zr +3r2 + * *'-"Q,(')=f'z +22x+32 t2 + "+n2x'-1'
aul l,(*\=r+ 12 +f + */' =t:-] (r*t) cemins ba:oq'
Gor[rduyu kimiP,G)= l,'(,')
--
v) y= ralrl=r"(rsg!,) r'=ff = *],,* r.
boraborlil dolrudu. Odur ki,
"o=[#)' - k' * r]r -lLl-)- r.' *' -
_ (z+ tH'r -(z + tH :x+ I -./*r + r -G-rf
(r-tf380. Ccmlar [90n dushr grxalln:
S, (r) = sin x + sin 2r +... + sinr,I,(,) = so6, * 2cos 2* + .,' +rrcostr.
Ilalli Birinci beraberliyin het tcrefini sir | -fe wna+ alane:
Sr(r)sial = sinxsiol + sh2rsiri +... + Un nr sir |,.t l( x 3x\
smrsrn- = tloo! t -*, ),.^ r lf 3x 5x\
rrn zrsrn - =_lru_ _*T ),
. r 1l zn-t h+t\3ttrrEsrlr, =- lcos 2 x-c{x Z r).
Bu qiymetJori yuxanda nezrs alaq:
s,cx.i=i(*r-"""?r,x 2n+l ,t ,t+l. cog- -oo8-r
5,1,1=!22=2,,2....2 sin; sh;
Axrnacr borabodiyin r - e gOm tliromesini tapaq:
( x 2r+l \lcd--COE-r I
4,(')=s.'G)=l '-.,'l=l'*r)
( 1 x 2n+l 2z+l \. x I x( r 2r+l )I _:srtr_+_ atn_rlsm___@s_l cos_-0o8_ x Ir\2--22--2)222\22)=
2 2r"-2
- "t2 I -oos2 f + (2, + l).in 2u + I stuI **, 2lf-1r"* I-'' 2 --- 2'- '--- 2 2 2 2-
zsm2 |2
- I + coe/E + 2zsh 4.rsin :. - 7rc - 2n+l x- - zsra'-+ in 3l,l
-
Isul-a ---,1 ") 1
2sin2I2
2sh2 I2
.zn+l .x itgzsE-rsln- -sul- -*r 2I
2
3t1, So = cfu +2ch2x + nchnt comi uqtln diisuE glxann.
Halll TrG)= shx + sh2, +... + rfuc comina baxaq. Bu
terofui .rt;-ye vuraq:
r,lx\stl= stu'snl+ shlx sh|+... + ru'shl,
go . ,1! = !( "53' - "1rl).2 2\ 2 2[
,nzr. ,t " =!("nsJ ' 3x\2 2\ i-"'1 I
. . t l( .2n-l . 2z-l \ttor' th - = - lch
- x - ch- x
)
Bu ifidaleri yuxanda yerine yazsaq, alanq:
zsur-z !2
csmin her
,,(* o ; = :lU oi - * r,). (* i - *+).
* n(.nut t, - *7,))= :(*!i, _ *i)Belelikb,
- 2n+1ch_r_ch -
r.(")= ---2
' 2' 2shl
2alanq:
s, =r) =l*t i*1". ]i-,r,f(m, i)'
Qalrynulor:
3E2, y=o5 +5q3*2 -x5. Cavab: l0a3.r-5xa.
36j. ., _q: !/- a+b' C.avab: aa+b
6u'r"6 ?!1r1, g,1*j.0 - r')'
gauayl -2!:2)-.(l-r+:r')-
cav*: -!:,.I4(l-r)r(l+r)4G*ti
iU. y=(x-a\r-b). Cavab: 2r-(a+6).3E5, y=1r+l[x+2)2(r+3)3.
Cayab: 2(r + 2Xx + 3)2(312 +l l, + 9).
3EA , =(rsina + cos @Xrcose - sira). Cavab: rsin2.a +cm2a.3t7. y =11+ runyt+ mtn).
Cavab. n4xn-t + {-l + 1m + n1r^+n-t).3re, y=0-x)(t.-x\2a_x3)3.
Cavab: - 1t - r;21t - 121t - 13y21l * 6, + l5x2 + l4J;.3Ee. ,=:-3.; *"*,_().3.*), u.o,39A. 2x
v= ---' ..l-r'391. I +r- 12
y = -------;.l-x+x'392 .,- r,- (1-r)"0+r)'
79
(2-x2X3-r3)' (l - x)'
- 3t5)ftl*u'
(1- r)P' Q* r1n
Cav-ab: -
, ='P (-l - ')q -- l+x
(,l*-D
396
397.
101.
EO
*u*. 12-6x-6? + 2x1-+ 5xa
0 + t)c+l
398
39'9.
tfz 2y=tx -Ir.
**o, ff,#tp - (q + t), - @ + q-r1r21 (rl*-s
y='*Ji*11i.
rlly=-+---+i=-x vx {/I
y=,W.
4N. y=o+xr$+l16lF.
r
/,,-rrr\-ln+m\x(:vab. . . =i-j-.
1n +m1 "'rln - x1n 1+ x)'
cavab: ,=t+-!-+-L (r, o).' z.lx tllxzCavab:
1ll-j-;T,-;sJ-"(r > o)'
*rur, J=*-! lr ro).3{r xVx
g^ru6, l*2]-.{l +,'
G*1,1-:).
caoot' " , Fl.lrl)
@2 -x\i
^ . 6+3r+Er2 *4f a274 +3r5l-avaD.
----=---'*(G - r)t(l + ,)' .
403.
405.
Cavab: - I3
(l + x2;i',[ * ,'<, * ,'I * ,* ,'l
4M. ,={r+{r+{r.
406407.
40t4t 9.
410.
411.
4IZ
413.
414.
115.
,=il;m.Cavat: l.:]-,, {*(,.,,r),
y=cf82x-2ei)x.y =12- x2)wx + 2:stnx
y = sin(cos 2 r) . oo€Gio 2r) .
/ = sin' rcos tu .
y = sm[sidsin r)] ,
Cavab: cos r . cos(sin r) . cos[sin(sin r)] .
sin2,sitr r'
6o""5, lul("*:tr{-lel*"1'z)
Cavab:t+2lr+4JxJx+Jx
EJrJr+ Jx r/r + r/r + Jr(r > o).
.ffi @l*o'l!*-rl{*t)
C,avab: - 2cos.(l + 2sinx).Cavab: 12 siu r.Carab:- sia2r.cor(oos 2r) .
Cavab: z sin z-l r . coo(z + l)r .
(x2 +ka lc =1,2,.-.).
(b3r
2sin" x
C.*U, - llT" (x+ktr; k=0,'r1,t2,...).2sinJ r
I- cG'rcavab: 'HJ q,*\J.ac r-tamtu).
msxLSrnr - rcoB r, -'--- - ----cqtr+ xsrn r
xxy=lt--clg:.tt
.'Cavab: --:^ (x+htt;
sin'r
crrab, '2 -.(oos r + roinx),
t = 0,r1,r2...).
416
417.
y=E -1,{,*!E '.Cavab: l+lg6x (**12k+Dl; r=0,t1, ")'
y=P'[;tt\;,[;t&.CaYa,b: --=j -
(r+et, t -tam edoddir)'3sin' r 3'r/c4:r
41& y=s&z!+@sJ!.
- l6"G?rCavab:
- ^a (r..Y; t -tamdr).. 1Zx 2
4Sm_ -a
419, y = sin[cos2(rs3r)] ,
Carab: - 3rg2x aec2 r'sin(zrEi3r) 'cos[cos 21tglx11 G+l+kr; t -tamdr)'
7.Cavzb'. -2o-"
122
423.
ll,=zE: c,',nas, -)zEi*'z\r,zy = e,lx\ -2x+2). Carab: r2et.' -[t- 12" ,-0*r)2*rrl"-,. canab: x2e-'sinr'
'l2 2 )424. y = ",
(r * "tg!).
c",nr,4@If! e*Zkr;23trl- ;
425. h3 si.nr + cosr'1'
ar osin6r - 6cos 6xY=e --J74F-
ry=qx sge +e"
Cavab'. "*11*e" 11+ """ 1
t2
t - tamdr).
l+h23 .LaV8Dl - _-slnr.
3tCavab:
y=,[7 *F ,*" b,.
42& ,=(;)'(:)'(r' @>0. b>o).
caab, y{u?-9:!) ,,,ol.'\. , x )429. y=*d *or" + oo* 1oro1.
C-anzbi ooroo-l * *a-loro ^o*oxoa'
.^2o.430. y =tst ,2. Cavab: 9be k2 12 , 1r ,r 0) ,
431. y = lofln(lnr)). Cavab: I (x>").rlnrHbr)
432 .v = ln0n20n3 r)). Cavrh: 6
" (r>e).
rlnr(h'r)133' ,= I
hn *.r) - I hrl * 12) - I' 2 4 ZQ+ x)
Carau: ---]-------=-, (r > -l).fl+r)"(l+r')
134, l, l-t Cavab: l- (El>l).t= Oa t- ra-l131 ,=--l-*16 'a cavab, -J- 1r+o).
' 4Q* 'a)
4 -l t '4 r{l +
")'436 ,=-l:n'4-tr3 cayab, i- U,E,' 2,!6 r.l3 + JZ
437. ,= I,t+r*.f ,l+r..8 (o<t<l).' l-k l-r l-t I - xr/t
43t y=..G+r-t"(t*Jr+t).
Cavab: =---]: rr > -tl.2(l+.,ix+l)
1j9' ,=loq"*{'*ty. c"-b, ,,|,
83
410.
441.
y='l,lo*[*])-,8;7 Cav"ab: tdx + rf,2 + t) .
y =,a2 @ * [ *-? ) - z[ + ] u@ + ^lt, ] ) * x.Cavab, ro21r* fri-9_
,-u =
t.Jr2 *o2 * llo,G *,ir2 * o2).' 2' 2
cavab: Ji + "2
.
t . Ji + xJbv=:ln-.' 2.td .ta - x-tb
y=be!. Cavab:=L (O<x-2trcr, t -tamfu).. -2 srnr
,="u(;-X) ce'ab:-r g-z*,1.i.r-tamtu).
y=lug2x+n,,nx.
Cavab: - "tglx (o<x-2br<r, [-trmdr).
Cavab: -l G *\) r.,k -t",r'fir\.
co6, /i +cos,Y=--==-+m.F--.-.' 2siz x ! sm;
or"ot, *.2' (o<r-zkr <r,t-tandrr).sinj x
1SL b*o***JF -i ",^* (o< El.Pl).Y=lna +bcasx
Jt'}"u'a+bcosx
y = 16113 r + 3ln 2 r+ 6hr+6) -x
11i
446
v_2+1r2 G +.rhr*J='x4r
Carab: - .! (0. r. t).x' ^ll - t'
| =ln
c,u,*,,fii (r.rE)
**, - t"3u'
1r, o).
, =)<, -'rt rJ l2 + r m(t + 3rfi
+ 12 ).2x
LAIAD: :l+Vl+x2
,={+.'(}.":)l_llItr+-+ln-
Cavab: - r r
[,-'r"1[,.,r[r.,,t;]
'=;i'1-#
y = x{lin(nx) - co(la.r)1.
Y =utgl-cosx'lntgr.
.x/=a$sE,.
l-xy =arce,os7'.
*2y=arctg-.o
IJ'! =
Trarcctg -y=Ji--de"G.
ca,au: $ (-ro).
456.
457.Carab: zsin4n r; (x > o).
Cavab: sinr.lo6r (O<x-2fu <1, t-tamdr).
Cavau: -!, 4tr.2y.tl4 - xa
461.
Cavab: ! (lr-ll <J7;.tll+2x-x'
c-avab::g--r @+o).x'+a'
Cavab: . !-- (r + 0).x'+2
Cavab: f rr>or.20 +,r) '
carab: -;ft-"*", (!rl).
462
463.
, = rurr.i, \E + orugJi - Ji.
orat: arcsinf{ tx>o).
y=r*dJ.rroorr.
465
4&
467.
46&
471.
172
carab 2tgtGb')=""st G +kt-k't,,.do,{l + cos'r
/ = arcsin(siB , - cls t) .
cavab: "i"i+P @<x-b<l' r-tamdr)'{slDzr
I)' = arE{o8-,
y = arcsi{siar).
y = arccos(cos 2 t) .
l.y = arcooc {l - r' .
1+'y -- arctg
L_ x.tr sin r + oos r)
v = arcctel ..-- | .' *\rmr-colr/t ( ln-b x\Y=dfi*"\'|;a"n)cavab: --]-..t+DCOS.r
. l'x2'Y = arcsn
r+7'IY=;;]G\'
y=arctg+:arcryG3).
), = b0 + sin2 r) - 2 sitrr ' 6cr8(siu r) '
Cavab' -+- ftl'D'lxlil - t
Cavab: sgnlcosr; G*?!in,k -tluodn)'
469.
470.
c""^b ffi (o<l'l<l).
cavau: J5 (r* l)
Cavab: t(x + f,
+ tr',t - tEmd$)
(a>b>0).
*116 _44* 1r+o).
c."*'t41+ ro
47;.
474.
475.
ca b. _, lr-- - (lxl<t).rit -rn ,r"co"'111
Canah: - 2ccr x' arcrg(sin r)
*."".+)
476
477.Y=Itr
t6
1t1.
2r{.Tt arc os iL(-r > I).
(0.Id<l),
(r>l).
Cavab. rllEq Gl<l)o - r2)i
l. r4-r2+t I JlY =
i2 " E;tj- - rTr*E 2,, .
corrb, o2 *!' - (rr-o).(x+a)(x'+b'\
l = |i "2 - 12 *lnrc"io I 1q 2 g) . Cavab:
l. (.r+l)2 I 2x-ly=-tr++-artls:.' 6 x2-t+l J3 " J3
Cavab: -=L 1x * - t; .f+lI , 12 + x.tD. +l LJ1
'= o,E't -*{ur-rfi*'8 ,\tcavat: -l . 6+g,
.p = x(arcsinr)2 * 2.ll] *r"in*-2r.Carab: y=(arcsinr)2 (ri . t).
arrcosr I , G,!l=' x 2 t*Jt-r2'
arccosrL:lvaD: - =----
x'l---: ln.r
y = *ctgr x- _r_m.
6"r"5; - t!t-
(r2 -t)iarcsior l. l-rv=:+-ln-' ,,J1- 12 2 t+x
179.
1M.
183.
44.
4t5.
1t6
482
491.
c","b, +1; (r.#),=fir-on n" ca'ab:
l2x5
A+ Y1212 '
(x < 1).
cavab: "-+ ftl<l)
CavaU: Jflt*ra
t& r_Ul ,. _ t +211iv = ln---_=_--:__ + 4 socg_6_.'
Jr *,G *1/r'
6"*5, -J *(l - r)Vx
xy = octE --------.: .
l+ Jl-ly=-ns+ @>o).
2ls-x'CavaU: $ (0<r<a).
tlm- x',=+,tt-;=+zarosinf .
c*ab: --i=-- (lx+ { < JI).tlt-zx-x:
,=i'm-i**Ey = octg(rgzx).
cavat: 7ift 1,*4J",t-o'0,,1't r't+@s r
, = r{:7." ff . i h L J+.,{= +arcsia.r.
c^".b, y=q -7ft,"[f; ,u.,
, = *o,n-i^(r+ x\-l@rctgxlz .
-2Cau&:
fiIarctgr
196
497.
498.
SN.
501.
s03.
5M;
y=t ( +,E+"b1.
y = ar"tg(x + ,ll+ ])
**, {{ 44*e.
Cavab:
cavab: i-.'h * "2'
cavan: --!-e- .
2(l+ x')/ sinasinr \
y = rcsltrl :-=- l.\l - co6 4oo8 r/*""5. sin ajgp(cc r _cos a) (cosr*cooa).
I -cc4aosr4ee. r.Jji-,$t .,17iv=-rn:+r@ctt--.' 4,13 ,lr2 +2 + xJiCav-ab: - . l=-- (0 < lrl . t).
(xa -l'1,lxt + 2
t xJi t . 'hJ -*Jlr,=:@.c19- ---ln-.' 2.12 " ,h**a l.lz Jt*ra * r.{2
F -----'=
x'll-x' 3 xJZy = .....---------- - --- arctlg ---:=.
l+x' lz ,h-x'(rl . tl
(t+xzf ,l1jy = oc-{rin 12 - *e ,2 ).
^ z*iol**rr2)uavab: ---r--:---J
/sin1zr2)
sqs. - fF,=oase'-n1l":,*r.
tg
(r.U=ffi' t=0.r,..).
ca"al, *-l .
e" +l
y = arqsin(Ein 12) + aIpco{oe l) .
Carab: z4sgnqoo 12; + sgnlsinr2l (U- f, o = 0,"r,. .).
/ = ,'*th'[mu(z arc'sia r) + sin(zr arcsin r)] .
C:lvafr: y = $et(d').cosz(arcs-D (!.1).' Jr. '2
y=arcc|+--l"'t?
C.avab: --------)-*'"" /*+f.,r-.*llffir = r"'(r." zlc).
z'*# to z. "^(
z*') . nt.* z[ ica'rab: ---33i8 ;{q-
50t' y=vx" *ro' * ox'cavab: 1'-rl'11+ar,r) t,'rd (1+ 6 onr)+fof n 4r *tr r)(r >0),
509.
510.
y=4i, Q>o>.
y=1lnx)':Pr.
512, y=logre.
513, , =1r1"6y*-1-.bh"r
s14' '=:k-4*1)
l"Cavab: r, (t - lnr) (r>0).
Cavab: - ]0oe,")2r
(r>0, r+l).Cavab: tl3x.
**b. _1 ir > o).sh" x
c"*u, 9I]1r' f x -21n2 r + xtnx.tnqnr)l (r >l),|lrr+l
il|. r .., lorlgz xI arcsln{sitr' ,) I
u =l
-j-=-----:
I' larcmslcos2 rl l
Csvat: "'.zl{]@;t
-6('h'.') * or"rrz' I sin' sg(oor) -[l+r' rroo6(c.fr) Lrrci!(ri!2 x)Jl + rin 2 r
_ coi,.rs(si! r) lI f,.r". ,=o.r..lrocoloo2
'1/l + o62 llJ I 2' )
90
5$. y = @cts(da>. Cav,p,: );.s16' ,=,.J-1-.). 6"*6' sF(lE) ir*o;.' ----\cb) d,xsI7. r=!,,rJ7lr,,n.iTrl;) (o<pr<a).
a a \t4.clrv?f,.. e+-bck
.b+eli
S1 TORS FUNI TYANTN, PANAMETRIK VO
QEY{T.ASKAR soIdLDO VERf.Td$ FUNKSIYALARINrORoMosl
1. Tts fuakiyanrn tbarrred y= /(x) funksiyzs'n"' tsrsi
r = 9Q) firnksiyasrdr. Tars funksiyanrn tdremsi,lrt='lGr
diistrru ile hesablanr.Z Poonaih dtlkla veilm$ funkiyattn th"rrlacl
t'=&) ,l=i$@'''nt
sistern tonlikleri firnksifanm paranetrik gehldo verilmcsidir' Bwsda 90va y(r) fimtsiyalan diferensiallenan fuksiyaladr ve d(r)*o{r'y funksiyasrnm r-a gtra t0nmasi bu halda
v', ='4xt
diisnru ile taprlr.3. Qeyi<Shar taldda vqiln* funkiyann fiadasl Ogpr
difersnsia[anan r=lr) firnksiy65r F(r,y)=o tmliyini odayirse, onda
y =y'(r) tirennsi
*at"t,rll=o
kimi tapillr. Bura& FG,y) r'in mnreH<ab fimksipstdu.
51& f +3y=v tenliyi ila teyin olunan v=v(,) firnksiyasmo
oldulunu mitelyen edin vs tbremssini tapm.
9l
flalli T$q b, bu tenliyi tidayen iki ,, = vr(r) ve vz = vz(x)
firoksiyalan var. Bele h,f13 +3Y=a, Y2l +3Y2=a
6d.oir. Bunlan t ref-are6 grxaq;
h3 - rz3 +?'(&- t/2)=ove va
b, - Yrl,Y? * YrY, + Yl + l)= o
yl +y1y2+y|+3>o btitEtr r-ler lig[n 6dmir. Odur ki, n =v, olmaltdr'
ir-' [u t Jivi efuym y = y(x) firnksiyasr \rar r/e yegan'dir' Indi onun
fitrermsini tapaq:
3Y2Y'+3Y'=1' /==+3y'+3
,= r(y) tars fiitbilasrnu va rq oblasnm tayin edin va onun
tdrcriasini taPtn:
519' Y=x+e'Eitlly',=t*"'' (--.v.*-),
/0)=l ''l+ e'
520, y2 =zpx qeyri - alkar frulsilashrn tiirsrnesini tapm'
EalE Z =2pay=P y*0."vSzt. *dgL =b,F * v' qeyri-aslor firnksiyasmm toromosini tapm'
xEolL
I .ryx -Y- | 2x + zrY-L--7 -7-=Fj;WtZ
rY:'-!=x:Wx;=(r-yly,'=**y = y,'=12 (, * y).
xz +yz xt+yz x-Y
522 a\ p=as (Arximcd oyrisi) verildikde Y,'-i @rn'flelli ,= pcose, y = psng olduEund& oweloe 1,= pwp
beraberliyxli, soffa y = psing tlrrabarliyini r -e nozarsn diferensiallayaq:
92
Ir=4"*,--o"ae*.
1,fr,,..,*frBu sistemdan ! ," ! ;-puq. Kiamer qayda$E g6re:&&
^=ly: -:y7=,, o =11,
-,"'E= rws+y'sins),Flnp pca6e I v pwPl
^,=F, r.l=y,"*p-.i,e,
lsrtrP / I
do L' do L' v'wo-sinaa=i=c(,sP+y 3*e, ;;= A= p
*rn=i, sine=l 614rr*, * * o*-"axnncr ifidalorinde yorine
yazsaq, alanq:dp _x+ yy' 4g =t/ -ydp.bp'
* = "* berabarliyinda burlan nezaro alaq:
x+w' xy'-v . a , avr+ W'=-rY - '-'Pp'PP1.\lG l, cD . ox+ovl_-yly =x+..!_, yx=, ..\P I P @-PY
52r. Paramdrik gekildo verilnig fi=;# (re(o,')) fltrksilasnn
t6romasini taprn.
Halli 4' =-asul, yi =t^t oldufuau bilerek melum dtstrtrdatr
istifi& eEch oda
' v,' Dcos, byx ===_=__cw4 -astt t a
diisturundan istifida edek. (hda aJanq ki,
, (3r+l), 3 .yr = _-___i= _= 5.(t-2\ r
@m.alli
(7,
524 Paramdrik gakiltb verilmis .lr=acos-' funksiyasron torsmasini[v =otio3 r
,,' =Y'1, - @"b1 t)''= -:"""'r'Ti = -* = -'s'+ 1aoos3 r)' 3ow' lsirrl oo3'
525 y=x+lr r, r>0 firnksiyasrmn toninin tdrsmosini tapn'
rralri verilen fiuksiya ii90n /'rx)=l+1'o ol* o"d"t
lllrU'<1D'= ,^=j='+t=l+,_/\u l+: ;
,,-
526 y=zri - xa , r > I firnksiyasnrn ttrsinin torrmosini tapm'
IIallL Voilen funksiya i4itn /'(r) = 4r - 4r3 = 4x(l - 12) < o olur'
Onda 7-t6';= fiffi tsrs firnksiyasrnrn torsmesi
g-rgY'= 1 = '
4r(t - 12)
CahgT lat:
Poamarih gat:ilda taila&S fu*dyals.n thra"tasi"i tqtl:527. (r="t *r2 t
lr="*'', t 1')Cavfr:. Y, =tt''talt + 4)
52& Jr = a(t - srntl . C.avab: y,[)'=a(l-co's'
5n. [x= sinzr
L=*"',530. [r=rcht
lY = btht
sst. [,=3r[:7[r=Jr-''r
Q*L+k4 t*7+tuk-lamdu)-
' =s1gL (x *2kr;k -tamfu).
Cavarb: yi =-t (o<r<l)'
Cawb: b-crtt
.
tn- -uc.*b,ii##
94
532. f t Cavab: yr' = sgl (0. 14. *-}lr=arcs,.JEzlr="""*.I[- Jr*r2
Qey iqka faaksfrabn yi warostal topu:533. ,2 ,2 _-;+-=l'
o" b'531. ,2 +zry- y2 =y,
s35- Ji*,ty=Ji (a>o) (penboh)
536. 2. 2 ?
t3 + yj = ot (asfioids)
537. r = a(l+ c.osg).
Cavab: -ae! 6+0, o*t!|.
Carat: l'=-fi.cavab: y'= 1iI2..
a-yCavab: r'=-rE.
Cavab: r'=1E.
y= tG) funtJ ralontn b^iinin tuorrwhl tepw:53t. y=slq.-
Carab: -ocy<r--, ,r' =
539.
1
Wt=th-
Caveb: -l<r<!,
y=7ft) diGrcmhlhnan tunls iya-snn qafikinin MT va MN toxumn venormalmn bnlikbri uyfun ohnqagagdak kimi oht
t-v=v'(x-\),v-v=-+w -,\
vBuEda x ve f toxunan ve
,I.,lt - y'
$. T ORorvr oNh{ HoNDosi MoNAs
nounalrn cari koodinathn, y' = f'Q) be bxuffna n6qbs inde t6ramenin
qiymatidir.20. Toxunan va namal pr gnlo I Pr- bxunanmm otIEcaEL Ptf -
notrnalm otuacag\ Mr -toxun r\ ,14v -normal ,8r=)" olduqlannl neztn
abaq ataf,dakr qiymatbri ahnq:
Pr=1.4.' ' -lYl'
*=iilr;F,PN=W1'
ttlt =l/[+ r''z .
30. Txunma nqbs hin r adius vektou ib torutnn o a mdakr bucaq'
Ogsr eyrinin tantiyi polyar koodinat sbtsminde ,=f(q\f -aItoxrmanr ib oM -radius vekbrunun toxuffna n6qbs inda ameb gBtitdiyi
brrcaq olasa, onda bucaq
rgp = 1 Ausunr ib hesablanrr.f510. 4 .n(-r;o)t b)B(2p); v)c(3;o). noqbbrinde v =G +tWz - '
ayrb ine gakilmig bxunan ve nomallann tanliyini yazn.
Ealti Ovveba frnks iyanm 6ntms ini tapaq:
y=(r+lrc-ri ,
y' =(g-,)l - JG *rxr-,) 3 =
,+t 3(3 - r)- (r + l) 8'4x=j,i_Y__'- .. 31/0-,P 3{/(3-,F 3t/(3-,)'?
Indi ;(- r;o); a(zr); c(l;o) noqobrindo turemenin qiymetini Epaq:
/(A)= ,+16=h=At yol=#=3=o; /'(c)=;*=..Toxunanm t nliyinde deyibnbri nazarc ahq:
o)Y-y=f'(A\x-x),v -()= ,2=1v .tl=?fzv=2x ,2:+2x -1,E-z=o
tl2 96
Nornalm tenliyi ,-,=-fr,@-,),3t_
r -o=-l(x +t); 2Y =Jl-2x -3Jr)lfzx+zr+3Ji =0.b) y - y= l'(BXx - x), r-3=0.(r-2),
y=j , y -y=--J'--1,-21f'(B)\' 't7@)=o oldutundan r-2=O nomalm bnliyi olur.
v) y -o=o.(x-3), r-o=-1G-3), r=0, r=:.511. f(r)=lu ayrbi ol oxu ila ha:r r bucaq altnda keslir.Halli lnx firnls iyas r Or oxu ib ro= 1 nOqbs inda kas b ir
,f(r=(lnr'=! ve /'(1)=loldutundan, f|)=lnx ftnks iyas mn qnfikim gskihni bxunanm bucaq
elEah ,ga = f'(l)=l olacaqdu. Bumdan a=I ahnq. yani, /(r)=tnrftnls iyas r Or oxu ib 45 dancelik btrcaq alhda kas! ir.
512. fi(x)= sitr vo /2(r) = sosl eyribri harsr bucaq alhoda kes S ir.IIaIIe Verihni eyrilarin kaslrne ndqbbrini tapmaq 09Un sinr=ccr
bnliyini hll ehok hzrndr. Bu bnliyin holli x= I n6tos ilir. Ahni r = Iohn n0qtsde hor iki ayriya gakibn toxunanm bwaq amsatmr [psg;
fi(x)=(sinr)'=c6a; :,'e)="*i=E,
, (r) = (c.sr)' = -sinf ; 1r'1'.1=-r " =-Q.' "'4 4 2'Onda ,(r)=sinr ve /2(x) = 9061
toxnnenhr uyBun ohraqJr( r\ Jiy= _t r__ I+_' 2i 4) 2
qaklinde ohcaqdr. Bu bxunanhr arasmdakr 9 brrafr
eyribrine (f,f) *or,r* **ro"
ve "=-9( ,-z\*E- 2\ 4) 2
, ,lJ, alo'=frffi^),r=l€1=t=*
q = 6rc\2J'. '
54. N'NKSiYANIN DiERENSiALI
1o . Futtks iYanat dilererc iah'
y = /(r) funls iYas mm attlmrAY= tGW+4a\
& =Ar kioi gOe 6rib bilisa, onda afimm bo5 hbsesi v fiurks iyos nm
dit lers hh adlarur vaay=e(*)e
kimi iao ohmur.
llrif '
n r*"o* mm sonlu tdramas inin varfulr' onun diferensblmm
Lig, irguo t r. ,oruri ve ham & knf ;artdir' odur ki'dY-- Y'b (l)
20. Funksiyan kiqi| ormnn qimat94i'ry?t!Funks iYa nm kig k artm rnm qimutbndirilmes i 09 0n
' lG+ Ar )- /('r)- /(r)Ar
dBturndan btiBde ohrnur'"';;;.jGt:-' -2x+r firnkivasr uetrn 1)a/(r);2) d(r) muevven edin
u, s)A,=i;'U)&=0J; v) Ar = 0'01 oHuqda onhfl m0qaybs edin'
igAt tl NoqEd. funks iya art'nmm bdftne gora:
a7(r)=7(+ar )-/(1)=(1+Ar f -z(t+a: )+t--13 +2-1=l+3Ar+3( Ar F+( Ar y-2-2Ar+l=
=a:+3( ax f+( ax f2)Noqbd. di$Ensblm brifine gtn:
d(t=f(t')Ar=Ar'ihdi ba m[qaybe edek:
a)A/(1)=s' ( '=r); d/(1)= I ( '=1);
b) A/(1) = 0,t31 (Ar=o'1); d(l)=oJ (ar=0'1);
, j 476) = o,roror (A r = 0'0 t); C(l) = o'ol (Ar = 0'01)'
511. r =LarceL (a * o) oriftten btifa& edarsk diGrcnshlmr tapm'
Hea dv=:+ i*=#1+-tt'
9t
sts. a(o') ap^.HaEi
545. y=ucE! dy=1.v
Halll
d[.,')=Q' * *'b=Q+ x)e'a.
3x2&=!- 4x2 - 3x6
, 1 / u\ u2 vfut - td" vdt-udv4v=-1tl' ,*(u.\'-\v) u2+u2 "2 ,2+u2
\,./517.
if,1G, -r,6-,e) ap,.
Halli
3x2&
sle. 1l*"'h'l t p*.dtarcoosr,
Hallil)-
d(arcsinr) - A -;*a ,"*;)-- -
I -=-''^lt - ,2
Futr&i iya utDn t difererc ialla avaz ebrak, oqribiqimatini taprr:sle.w.fuA y=tJ" funksiyas mr goflink. Bu f.rnls iyanm Erornas inin .r=l
noq6s hda qiymatini apaq:
. t .'y =_.r-i. yI,)=1.
(Ar=0.02) kigik atum d0strnrna gdra
Vr *a,={r*Ja,=loore.
J50. sin 290.
Eolli y=sinr funksiyrmr g0fltok. Bu fur:tis iyanm ttinmes inin
qIyIIFlm I r= - -Oa BPaQ.
99
/P1r,-u'-"1="A- ,2 -rzrs -greL. -3r2(t-4r2 -3rib
,'="*., ,(2)=*Onda kiik artnn dtb urruna gdE t Or=-# )
"i"zeo =,i,[f - #)","T' * k=1['- #) =.'"'
Qolqrrult:
1
xt - E-al
v = - lnl----].' 2a lx+al
cavab: -$ (r+o).u
c,na' ,fi (E*l,ll.
&Cavab: ----.
'J *2 *o
crat:ffiar'\a -r
Cavab: xsinrd.
Cavab: -{ (r * o).
s"*6. ?:$a, 1,, o).
gu*6' --a- (rl<r;.
(r- x21i
."*6, _{ @lcr).
c"*b, -L[*rr- tin').zrz\ x )
Cwrb: - ctgt (t*kr, k'tzrr, adeddt).
Cavtb: vwdu + uwdv + uvdw .
, vdu -2udvCavab:
-
Y*0.v
100
555.
ss6
557.
sjt
55r' r=r"1..v7.,1.
-x, = arcsln- (4 + u).
o
d(sinr - roos.r) '
,/r)dl -l'.fhr\\.Jx./III
IJr-,'Jdln0 - 12).
d /sin.r'\--------l
- l.
d(xz)\ x )d(sinx)
.
d(cosx)
Y = uvU.
uy=_,
559.
560,
561,
562.
s63.
561. I ^ udu+vdvy = _Tf_:_ . L;alab: _ _-__________ .
1'r u +Y (u2 +v21i
565' y=nJu'1+n'z . cwaa:!7$ 1u2 +v2 >01.rt' + v"
566 {#L.;"f"(;.;)) ""*6' ${'* L+kr'L-t-,u,r)
s67. ot.l] *ll. x&uavao: ffi.56t. / r) &
al arccos- l. uavaD: _- (lrl>l).\ l,l, x1lx'-r
569' ffi catab: -tg2x(x*l+hx,k-m6i
S 5. ytrKsaK ToRTbr,irOnauevo DiERENSiAL
t0. /(r) funls iyss r z-da6 di€erhlbmn firnh iya olduqda
rt,\a=t'*"G4' G=D..);
a" 7 = aQ"-, f) b=2,t,..)dtbtufun do!rudur.
Ogar r serbest dayiandisa42y= 43y= -..=9.
Bu halda
a'1 = 1@16,yd&turu dotnrdur.
20. *as dBtudan
t.@f) =o'ro'o (,,0);z. Gurl')=tir,(r*f);
s t" #)="*(,.?)'a.Q^f) =,1, - ry. ln - n + l,,,-, ;
t0r
s. 6",;t'l = (-!r-!:4.
lo . I-eybnis d* drufunlsiyahdrna, onda
dlb trru do!rudur.
HaUi
tro.r=ff v'='!.
HaUi
,=* - i=_r_x2+x2 =(,_,,I1.. t_x. (r_rrF
,' = -)l- *li 1- z;1= il1-,'l';s71. y=h.f(x) r'=?.flalll
.,.-f:o .,._r..f -r,2Y=-7(;' '=- r' '
T toq ki, u=,/x\, 't = /(x) *i dafa dirtrers ialltnan finlcs ialodr '
y' -i tapm:
572. Y= ttz Y' =2 '
Ogsr a ve, z-dsB d iE rens bllanan
1-1") = f ";,t'),G-D,-0
y' =2uu', y" =2u'2 * 2*' = ZQr'' * ur').
573. y=v' (g>O) y'=?.
IhllL lny =vlnu,L=r'1nu*4- = r'=u'(' rnu*!!),y.
r =b"llt u' *4:)* " (" n":! .'1,)=
-_,,( u,nu * !\' * ,,f ," ro o * 13;IJ * {].\ a./ \ z' u)
ve. y = 1$'l r' ='t, y' ='t.
Halliy'=2xf', y' = 2I' + ttz 1' = /1t' + 2x271
y" =2b,1' +4,f'-ei 7')==+,(1' +zy *2,'7')=+$7'*2,'f')uunaa 7'(,']),7'(r'),.rl,') kimi oxunmabd rr.
575. y = e' funks iyas r 09fln
a) r-seftestdey$an olduqda; b) r-aralq aqument oHuqda d2y-itapn.
HoIIL
at ay=aQ')=e'at; a'y=&'a)=e'(a,[.
D dy=d(r)--te; d'?y=dp'a')="'(&)' *"'a".576 y =.,R' r - i sefuest dyilen gebul edaok dly-iapm.
HaW
ara(t:.")=**.
^h*? -- l'-d,,=dl -+=e)=
"' " f;?(a), =, (&t,,.
' (Jt _,, ) t + x, (r *,rf,578. r=w ! i'a v-ni 2 - da6 diErcrsblhrun fimb iyahr hesab
edarsk d2y -i hpm.
HaIIL d1 = d(n)= vdu + udr ,
d2 y = d(vdu + udt) = dadv + vd2 tt + ddv + adz v = 2dtdv + "dz
u + udl v .
579. y=qrs1g!, 42Y=t,v
Halli( u\ I -/ri\ v'/ vdt'udv vdtt - udv
dy=dl orctS- l= ------=4 - l= -;----;'- ,-= ) --;-,- \ -v) t+r". \Y) u'+v' v' u'+v'
v'
,z .. - I ua, - Q'\ -b2 * u2b\a, - ,a"\ -d-y_ut z-2)=_6q_
103
2lva/-udvYdt+vdv\ vd2u-rd2v {""1a"f - u2 a'a' * t'o^ - ""@f)=-T,-,,f-=--i;T-- tu\E-x va y poonetk Sakilde verildikb *l rz,t'*z bramabrhi tap' '
580, x =2l - t2, y=3t -t3H.lll
., = y',
-3-3t2 -3.(t-rX+r)=1u*n." ,i 2-l 2 l-t 2'
,i =*[it'.'!= illP=i *=i*=ql..,r,rr) -1
t, = *lin)=lP =ifr,= i f; t'. o
StI. x2 + y2 =25. y=lr) qeyri+;kar firnls iyasl Wtrr:. vx'*,vitdnmobri tapm.
trbZt Veribn tonliyi r -e nozeren diGrcmhlbyaq. Onda ahnq:
2'+2tf"r =O -t'=-j'Axrmcr ifrdani yenidan diCrcm blhyaq:
l,-\ ( ,),,. =-"1il =-t-:r' =-'-\- ,)=-r'*l --u!r2 - b y2-' l,
- y3 yt'
.,.. -(- x1' -7sY2Y'' =75.y'a =-75.
x,.
"'=[7],=l'- v" v"7
5t2. t=L i8)=t .
Ilalli Yeibn frnls iyanr elverili gakiha yazaq. Ona g6ra sueb 1
ehva edak va gxaq:12 -t+1, = :____::_: = _( + r) + (r _ x)-t
Almmq ifidaniardrcrl di6 remhlhyaq:
.yl =-r+r.(1-r)i'?; ri, =r.2.(r-,)-3; y:, =1.2.3.(t-rYr04
,l?=r(r-') "=# G-D.
JtJ. ,=P-,, ,('*)=,.IIaA Veribn funks iyam e lve r! li q akilda yazaq:
,rlly = -J-Lt - Jt -, =z( -'f; - ( -,f ;
Almmp ifrdani ardrctl diGrcrs blhyaq. Onda ahnq:ll
)t,=(t-xY, +(-,]-r,1. 5 r 3
y.,=;(t_,f , *;(t_rfr,
*=| ]o-'ri,.) ta-,'ri.
/(loo) =fft-,rT .fft-''rY =#,.6 o.r.
5E1. y = azsb , l('o) - t"pr.
Ilallt Leybr,b dtlstrrundan btiAda edek: ,=12,t=eb obun. Adul6nrbbri bpaq:
u' =)y, y'=), y'=Q,
v' =2eL ,v' = 22 e2t ,v' = 23 eb ,..,v(?o) =2m . s2' .
Leybnb dibturuna gOn ahnq:
5z"z,f0) = r{o)"{zo) * "ioo,,fe)
* 4,,,y(rr) + cio,,.v(r, =
=a2.220"2t +zo.2x-zt9 -e2, + 29J2 .2.2r8."2, -220."b .12 +
+20x.220 .e2' +zo.'tr.rte ."2' =z2olrz +zor+gsbb.
5E5. ), = rsrtr, ,(ro)=e.IIaAL t = \y = sl:c ib ian edak:
u' =l,v' =clt, u'=0,v'= slt.
kybnb d(b trrundan bti9da edak:
(o/ll)(loo) =,,ov0o0) * "i*r,r(r) *",2*r'r(ee) *... =
= x. slrr + I 00. 1. cix = xsla + lOOcla.105
,-i sarbast dayrsan hes ab edorak, gds brilzn
dirtrers iallaytn:
586.y=7s, dt y=2HaEi
dy = 5xa &, d2 y --20x3 (af ,a3 v = aox216rf ,
da y = rzox(&)a, a5 v = tzo{*f .
587. v= ^L yb\ =t' x'-3x+2EaAi
x2 _3x+2=os t=r* f- =? o =,,
Verilen firnls iyam a he ri; Ii gakilda yazaq:
tor tibdan
x2=2 '
58tIIeIIL
589.
Ealli
funls iyanr
b tifide edaoh verilan
r1l/=GrTIt] veYa Y= -x .-- -,
''=-f4.ri'))' G-2I G-r)',-
,r"r=1_rra[ffi #,.]y=sin2 ,, ,(') -i ,np..
t=lQ-*"zi=l-)"*xY'=sin2x'Y' = Zcrs2't'
,' = -22 sit2x'
ul'l=-2n-t"or 2raA).' \ 2)1, = 51ar, ,(,)- 16p6.
51q3 .r= 15i1;s- 1r1n3r diblurundan44
3. IJ' =:sillr-:sitr3,
kimi yazaq. eger sin x firnls iyas mm (rio 'f') = ,inf, * {l ao*rrru' \. 2)nezen ahaq, onda ve ribn frnh iyanm z - c i ta rtib t0nrnm ini tapa nq:
,(,) = f ,i,(, * 4) - l. r,.a(r-* * 4).' 1 \ 2) 4 \ 2'590. Y = 5in qYsilr 6r, Y[')-i APrn.
Halll Ovtaba sinrs hon has ilini cema gatirak:l.
r =;[cos(a -aF - cosG + a)rl
Bundansona (*rrI =.oJr* 4) do.turunauo btifrde edak. onda\ 2)
,at = L{G - t) oosla - oV - if- t".rf *.lt . rf,. ?] }abrq.
597. y = si112 qx.pssDx ,1'G)-i60..HaAL Yeibra funtsiyanr alveri li gakiHe yazaq:
, = l(t - "*2,o)"orD, = *tD'
- l*r**rD, ='2'22= !9L-DI _ 1 1"*1o * 6)r + cos(o _ 6!] .24'
Bundan sonm (*rrld=*r(r*4) aw*.na"n b tifida edek. onda
ahnq:
t \ = lo' *"(t * T) - i{a, -'r *ltu - tv,' il-- 12, * af -,[12, * r), - +l]L', 2))
592. y =siaa v',sssar , r(")-i 6r..Ilalli \ eihn frnk iyam e tver$ li gakilda yazaq:
y= sin{ r +cos4, = (in,, * *r, rf - zsin2 xcos2.r =
= r - ]'ta' z, = r - |{r- cos+') = 111*r4'
,(') boaften (.*rI') = *J, + 4) dttn .od., b tifrde edak. onda\ 2)
alanq: 107
,t t = L + *s(+, + !) = "' {*. ?)
593- y =xcosc v(') -i gpm'
tt ai rcyV"l '
65x1run'lan ve kc inrsun n-ci 6rtib tdr:ms
arst."a"J'tqrb edsk' om g6n "=r'v=cos4{
ils igata edak'
u'=l'. u'=0,
,(,) = 1oo56gf,) = a, *"(* - ?),
,o) ='o' "4* * TJ- -'-''*[* - tt-ur) =
='a "{* - r)- *'-' "in(^ * i)'
591. y=e'cosr, Y(") -i t"Prn'
Holtt Y eribnfunls iyanr ,="' '"" T-" =Lr"t*ix *I"{r-i! geklinde
QalqTsb:
tl1 lesdbbYw:
59s. y=J;i. an*b,{'*''t).(1+ 12)2
flga !=e-'2 . cavab:. y=22-'2 12,2 '11.
597. y=ts).. Cavab: 4* (r*2k)1 ,,t=0,*r,..).
SgE. !=(t+ rz)dtctgx- C*ab: l=z;+2octv'
yazaq. T6rameni hesabhmaq 0qu' ('-f) = ""- ditstunrndan istifida
e&k Onda ahnq:' rt'l=jftr+i)rr(r*i]' *6-;Y'-(r-')']=
i ,-/*ry) 1 n\'l
= ll 1lzl; r I I *1r5Y-r,'-'1"'n )l=l,') "' *'(' *Y)'
'zl' ]
108
599, arcsinr9=:.'
^h''2600. y=rlat. Cavab:
1 lrrO;.r
601. y={sin(lu)+cos(hr)1. Carab: - ?sin(nr) (r>0).
602. .y = esin' cd(sinr) olduqda Xo), /'(o), /'(0) hesabhym.
Cavab: y1o;=1 y1o)= lb /'(o)=0.,r=p<x\ ve v =yt(x) fulobr Xbcibrfr frenobrl obn
fuabpob oWuqda f J lwablayu:603. /=hi.
"^oo. au'-_u'Z _-'-_t'2 1yu>o),nrz")
6u' rJ7;7"ro'.
@2 + t2 \w' - -) !(u'v - w)2 1r2+ v2 > 01.
(u2+t21iftdaq hl, fQ) Agturl rarfr tbtta I obrt Jurtk Udt. y',y'A
hadlqn:505. -/l),=1\;)'
.o-v,,,, = ) r{l). ir(:)',= r,(i) :,{:)-i,(i)
606 y= I@,\Cawb: Y' =eb 1'1"s1+ e' f'1e');
y' = "3'
f'(e') + 3eb f (l\ + e' l'(e'').607. y= f(bx)
cwaa: f = )V'gn:) -,f'gnr)l;
f = )V' <n t - s l' (h x) + 2 f on il.
a"o6r -{-* (l * z'2)t:tio'
44.9.(l- x.\. (t_ x2\i
cawbz2ffa, (rro).
cavat: ,,fo+urf -:y' .
60t h.rx
609. t=t,
Ifutaq *l u v, v firrlstlal@t x oqsrzrdrrbrr actb fttncl btrlbItenwtobn lukpab obse a2yl lstblcym:
6lll y =udvn (m,n=cowt).
Cavab: ,'-2r*2 fr{ n -l)v2b2 + 2mtutddt + 4n-tltrzdv2l+ u"1^na2u + ,ruazv1}
6II,612,
Cavab:
613.
Cavab:
611.
Cavrb:
!= at (a>o\ Cavrb: o' lna1fu2 ha + dzu)
Y=bJi +v2 '
lp,' - u'br2 - *rua,a, * b2 - i bn' * b' * 12 W2 u * n a' ul
b'*u'Y,I = 4c6r, / = tin,
I =-J " ; y. =- 3.*"t, (r + tz, t -bmdr)sin't t sm' t
r=a(t -sinr), )r = d(l- msr).,
l=--) .; rr=- :1 , (r*2rr, i-tamdr)- c"drol -
4"?"bri
61fr' t=et5a15,t, y=etsint.
(r2+12ro).
616 x= l'O, y=f'Q\- l(,)
cawb: r,=7[, ,=-ffi 0'to*o).
Qcytt4t*c p*ilb va:ft;/ry y=y{r\ lrlt*sfa,t OgAn y"', !,2 ,
\r lcsdbYm:617' y2 =2p cawb: y,' =2, ,,; =-i, ,.,, =ri51& ,2 _ry*y2 =1
C*ab: yr' =b--l , y-r" = 6 ' 54x
x-zy - <r- 2rt'r,t = (r-2'45'
ll0
' n=#;ffi (r+!+kx, t=o'tr...).
Qeyrlqks Sa*tu ver&llr$ y=y$) faab|.ost fibtlueb@n:619. yz +2tty = a4
octg!.t (r, 0).
621, y=x1u -t\2(t +3)3
Cavab: y(6) =4.r, y(7) =o.622. av=-
T
,(6) =t y,s ,Q) =7 .
Y0o) = t
/00) = ?
c*ar: ti =fi ,,; =#brt+yz)2 +zx4(t-y2tl
620.,1,2 * f =*
x-y Q-vfAS aqdah runls ilsbut vetbn btdbhn lfranobrti ks abbyn:
cavab: f ---@ff!2 {,*o).
623. t=J;ca'ab: ),(lo) =--ioll!-=, (r > o).
621, e,x
Cavab: y(ro) =r' f ,-rl'1io, handaku /i0 = 10 ' 9 ' ... ' (11 - i) ve 400 =r.' i:-t xt +t '
625. .y=rlnr yG\ =z
Cavab: y(s) = -1 1.r > 01.x"
626 lnx y@ =lr
Cavab: y(5) =4-Yu, Qro).r' r-
627, y= x2 sit2x y{50) ='l
Cavab: y(so) = 250(-x2 sin 2r + 5or"*2, + l125sinzr1.
2
I ll
628. .,:_l9lI_ r'=?' 1^h-:,.
cawb: f =!!:Et:A,i'a, -!o:1\4*,r, (,-i).(1- 3r)3 (1 - lx) a
629. , = sin.xsin2.rsin 3x ,00) =,Cavab: y(10) = --2t sh2x - 2tt sin 4x + 2t' 310 sin 6r.630. !=e, oosx ylv =2
Cawb. ylv = -4er cosr '631.
-y = sin2 rlnr yG) =t
cavab: y(6) = -!e . [5 - f . T),r, - [$ - S. # + 32 h,)cosz,.
632. ... I d3y=?y= rg"rsg' - --1:-6' (r, o ).
8x'Jx633. !=rcoszx dro y =?
Cavab: - 10241rcos2r + 5sin2r)dlo.631. y = er lnx du y =lcavab: e'fln.r+ 1- 7 * I - 7l*n .xxlx)x"635, ! = cos2r.cta d6y=?
Cavab:8sinxstr46.huq ki, u funk iyoat x oqro*ftindan oth cltrl brtib frraficsl ohn
lank la obsa gAsbtlbn bt tibbn tbanobr i laabbym:$a !=r2 d'oy=?
Cavab: htdt\t +20dtd9 u + g\d2ud6u + 240d3ud1 u + 420d4ud6tt +252(d2u12 .
637, !=et d4y=?
Cavtb; e" 1du4 +6du2d2u + 4&td3u +3d2u2 +dau).
63& )=hu d3y=?
^ . 2du2 3dtd2u d3utJa,mD: ---t-- 2-+;.
112
,t@)J fusabtoytn:
639. @+b' u+d
610. l' ,lt=u'
611. x' ltli'
612. y = cas2 x.
613, .y = cm3 .r.
(t11. -y = sinc sin rr.
ca,.b. eD"-rtdc"-r (d - bc)
(q + On*lgavab: lt,:12+ G.!)
(l-2x) 2
Cavab. (-l)"+l .1 4.....(3t -5X3z+2r)
n+\
3z(l + r) 3
(n>2; t * -1).
617.
61E.
Cawb: z"-t osl?-x +ff).
Cavau: Jc.s<,
+ f 1 * ! "*t, * ff1.Cavab:
Cavab: (-t;'e-i[rz - 2{a-1x + (n - t){n - 2)]
Cavab: ,,{l+ i r-1r dt!=Q:j:II[, r]r' x*+t )
cavat: ",21"6(,*t").
cavab: d#[(a + 6r)' + (-r;,-r1" - r'f r (H. |,D.
113
c," b, @
!)' *{<" - o,. ryt]-e:t"$t,. w. ffif615. -y= sinccosDr.
y=(? +2x+2)e-x.
e'x
.y = c'sin r.
6 6v'6, (ot,1,l<, - u, * tz, lf- e:U-,r,,1t,.,. a> - f l] .
, = x2 sma . Cavab:
cavab: o"[,2 - fff""(^..2) - r,a" \*. ry).
650. o+bxv=|lI-' a-bt
s6. ROLL, LAQRANJ, KOSiTEOREifl',il'i
ro.Roll Eceml aFr: 1) /G) firnls iyasr [4,b]-& byin olunub ve
kas iknezdise;,:l 1r\-i-1r,07-a^ sonhr toornasi vasa; 3) J\')= lb) ohsa' onda eb
c e(4.6) n6qbsi var ki, /'(c)=o dofrudur''zo.Iaqranj
teqeml Ogan 1) /('x) funks iyasr [a'D]-& toyin ohmub va
fre. ifr"raio", Zl (4,6) anh[nda /(x) firnb ias ntr sonlu t$omesi vasa'
on&7@)- 1Q\= 7Q\t - o) (a < c < t)
d{buru (sonlu 6qbr) doPnrdur'--l;. iri teqini. oear' l) /('r) vs s(r) fimls.ivahn [a'r] paaas n<la
t yln oi*ru ve kssilrnszdisa; 'z)
tG) w g(r) ftnks iyahnnm (a'd)
anhlmda sonhr Oramabri vasa, g('t)+ 0 olduq/ja
t@-A")=t0. b<c<blg(a)-eG) e'(') '
dlbtnru dogrudur--7i1. - fta=t-rXr-zX,-l) tuolsivasr 09un Roll teorcminm
dolnrluf,unu YoxhYn.*"ii;fr; iG)
-nrr.ryas' [r:] parvasnoa 0]!5-mlttaSanar ve
,=," ,=Zx=i' nOqobrindo s't''"'ge"itit' Beblikb [z] ve [z:]
parq-ahnnda Roll teoreminin qertini Odeyir' onda [U) ryrv1r-aa]inaa eU
il;; il fiurksiva bmin noqeda srfia beobt olur' Yeni /'G)=0'
inOi n of* iyao, Oifttmhlhyaq va sfin baraberedek'
f'G)=G - zX'- O+(x - r[r - a)+ (r - t)(r - z)=o'
x2 -5x+6+x2 '4x+3+12 -1x+2=0 '3.r2 -12x+ l1= 0 '
Bu Enlii tEll etnakb axtanhn noqtoni tapq:'E-tt
,r=l-tl' xr=la1-! ' l<c1<2' 2<c2<3'
652' Eb e=oQ,Lx ) tunts iYas I taPm ki
l(t + Lx )- f(xl=L{'(x + 0 Lx ) (o <e < t) obun:
$ IQ)=*'z+b+c (a+o);
b) /(r)= 13;
v) r(r)= !:x
4
q) IQ\="'.IIefia) a(x+Ax )'z+6(x+Ar )+c-a2 - bt -c =Ari::(r +aAr)+ 61.
eundan a=|;
(-r>O &>0).
o, (, + axf - ,3 = x3 + 3r2a, + u(arf + (axf -' -rr = 3r2a, + 3r(arf + (arf =la,(r+ruf .
,/ra, ' -l(a.r)' * ,' -,Buadan d= Y J
,t I -l=- ,& .=- & -. B,,o,a,ne=L( .fE-,'1.'x+& x r(r+Ar) ,G*fut| Ar(I x )
q) ,*n& -r' = axe"tu, Bunaan a=!rne&-l .Lt Ar
653. lo,2l pargas mda tayin edilm! .f (x)= 4r3 -5x2 + x- 2 funlsiyasr
0qtin Laqlanj dltsErundan 6-nin qiymatini TEpltraIL
IIaZi laqmnjm sonlu artrm dtb orrundan btifida edak:
f(b\- f(a)=f(€)lb-d).Aydndrki
IQ\= f(2)=12, f(a)= f(0)=-2,f'(x)=t2x2 -tot+1, !'(r) =1212 -lo€ +1.
Atnan qiymetb ri laqanj dEstrrunda yazsaq12-(-2)=f'G).2.
Buradan baf'({)=z
olur. Bagq sdzb tz€| -loq +t=7 .Bu tanlikdan
- s+ Js|'12
+4L edadi t0,21 paryasma daxil olnadrlr 09tn, onu goflirmurilk.
65r'. bbatedin ki, oca<8, p>1 olduqda
poP-t 1b - o1< bP -qP < pbP-r(b- o)
bmbos izliyi do[rudur.
fraUL x>0, p>t olduqda fG)=xP, r>0 fuoh iyas rna ba:oq. Bu
funls iya [a,6] pa9asmda [aqanj tsoremini trtbiq edsk. /(x)= rP
I15
funh iyas r la,b'l trurgasm& kas iftnezdir ve (a,b) inervalmdadiErrshlhnandrr. Iaqranj boremine gbo eb € e(a,b) varki,
bP -aP' = P'|P-tbb-o
be oberliyi do!rud :ur. aP -r . € P -r . bp-l olduEundan
ve ya
beabes izliyini ahnq.655.
*r-t .bP-=_!P . ,6n-t
paP-l1b- a1<bP - on . o5o-r1b - a1
, 0(x<l
funks iyas r flgfln to,2] papasnda, sonlu arum d0sturunda f-nin qiymetinitapm.
Helli !(x) firnls iyas r r=1 noqtso inde kas itnezdir. Do[rudan da,
/0+O;=,timo!=t,t--2
/(r-0)=rlito?=1, /(r)=l
oldulundan/(1+ 0) = /(l - 0)=/(1) =1
benberliyi ddanir. Digar taoften /(r) frnk iyas t (0,2) intervalmda
di€rers blhnandrr vs
l- r, 0<x<l olduqda
/1')=1-+, l<r<2 olduqda.lt'
Onda Laqranjm sonlu aItIIn d{buruna gotr heg olmazsa bir f e(0,2)
n.hbsi var ki, !(2)- i(o)=21'({) benberliyini ve yaxurl f@=-:bnliyini 6dayer. /'(r) ftnl$ iyas I ki dlbuda ifida olundufmdan bu
tonliyin heli 0<.r<l olduqda €=!, t.r<z oHuqda bo 6=Jl olar.2
lr6
656. fG)=14 x€[-I,+U fi,nksiasr Rolt boreminin Seft]rriniddayirni?
HeA "(r) = lrl frnksiyasr [-U] paryEsnda kes itlez vo uchdabeober qiymotbr ar,. Iakin porya daxilinda b.En n0q6brde .rsrroninvarlrtr gerti pozulur. Mesabn, r =0 noqtas irda n o.to ly"rr,
- Oo.", iyoxdur,lemli, /G)=0 tanliyini trmin edan 6 nOqas i yoidur.
d57. Bbotedin ki, egorheqli ermalh z datacalipn(x)
= ao5! a or*n-t +...+a, (ag*O)goxhadlb inin b0flln k0kbri bqqitbe, onda r,,1r;, t, 1xy,..., e!"_\ 1a.Sadp tEorcbrin da k0kbri bqiqilir.
HeA p,(r)=Q 5sS;i tonliyinin k0kbdni xt,*z,-,xzftesab edirik klb0flln k0kbr sadadir) Onda ,"fr.ju,danceI goxbdlb ini
agxn + alxbl + ... + o, = a|J(t - r1)\x - t2)-.(x - xr)geklinda gEstrn bibrft. Har bir [r1,r1*1] (t=1,2,3...,2_l) paryas nda.v = p"(r) funh Vas ma Roll EoEmini btbh e&k. Onda Roll tsorminegOn her bir (rp,r1.,1) intewafunda eb 6r n0qbs i var ki, r,,15ry=g
tl*rry, ddanacckdir. p,'(r) ^.nksiyasr (a-l) daecali gox6dlioHulundan ve [x1,x1*1] paryahnnm sayt ,r_l oldu[undan aydmdr ki,P, (r) = 0 bnliyinin b0t{to kdkbri heqilitir ve
x1 <(1< x2 <(2 < t3 <...< xa-t <€n-t < xn,
h,h*.e ,-r 3dadbri pr'(r) goxledlbinin kOkbri oHupunaqn be1 girl1r4*i Q=t23,...,n-z) paryas nda y,=pnG) frnhiyas na RolItsoremini ta6il ebek, eb 4i e(€t,i+i n6qb bd taprhcaq ka rj Ot)=o(i =123--n- 2) tnnberliyi 6demcakdir. r,"1r; frnh iyasr (z_2) daocaligoxhedli oldu(uatan va l|i,lt+ll pa4€hnnm sayr
" _ i oHugunln mm
ede bibrft k! el G1=s rrntiyinin b0t0n k6kbri haqilidir.Prces i davam edisak, (z- t)+i addnnda
rf-D 61= anax + q(n - t1t
oldulundan pb-r) 61=0 onliinin yegane .,=--n 6rq; lukttnll alnq.
ib igaa edekk\ p,(x) ,t
tt7
Qalgnnlu:
6sE. t-r,tl pargas nda I@)=2-{7 frrn}s iyasma a)Roll Eoremini' b)
I-aqranjursonlu artm teotemini htbiq etnek- olnrmt?Cavab:ohmz.
lr659. /(r)=l"i';''*'' fuoh iy*'" [0,1] pargas nd8 Roll Eotemini
[o' x=o
bbiq etmek ohtrnr?Cavab:ohr
lx +3' xr-l o'uuo* fi.ok' iut r Roll teoreminin gefibrini
660. le't=L4 , .r > _l o*luqda
Odayinni?--'i7i- rrqr"n;, sonlu artrm dlburundan b tifida e&rok a5af,dakt
beobes izlkLrin do[niluEunu bbat etueli:**-- "ffi;r-ti.r]l=F,
- al, x1 ve 12 ixtiyari heqiqis&ddil
b) arctgr2-arctgxl312-\' x2> 11)
662. lj\ll paEas nda 7ir;=32 vo g(-r) = 13 funls iya lanna Koqi
tsoremini ham rsebebe t'tbiq etsnek oltnaz?
563. fi't=I funts iyas I [-12] paqasmda taqraoj eoremininr
geilbrioi O&Yinni?
[.r + I' r<0 otduqd86H- f(I\=4 - -
''* fi-kiy^r [-l'tl paqasnda Roll--" "''-' l"r, .r>l olduqda
tsoreminin gartbrini 6daY irmi?
66s. lr,l=ilxar'J tunksivasr l-i,}] *"*'" taqnnj
eoreminin gsrtla rini odaYirmi?
-12667. f c\= e, u" p1r; = -l funksiahn [-2.2] p6$as mda KosiI+r'
Eoaminin 9e rtb rini 6daY irmi?
ut
s 7. FUNKSiYANTN AnTMAST VO AZALMAST
t0.Funks iann sbnastva @ahnasL 7(r) frnlsiasr [a,6] parpas mdaaftndrr (azahndr), egor
tlil> tGJ (a! x1 < 12 s D)
(ve ya 7(r2)< r1 . ;2 < J(41 (a s x1 < t2 s b))Oger di&rercbllamn /(r) funks iyas r [o,a] pa4as nda artrna
lazaLsa) onda
f'(x)>ob<x<b) (ve yay'(r)<o (a < r< D))
20. fi/nksiyqnm conasr (aahnct) iigiin kafi S*t. Oger,rG) firnk iyasr
[a,]] paaas mda kosiLnezdise va onun daxilinda mib bat (msnfi) 7,(r)idnmas ivara, onda /G) ftnks ias r [a,6] paryas nda arbndrr (ezqhndr]
Funks iotm otaatta aalma oahqlo mt laprn.
66t. y=)1-a-12.Halli Yeibn firnb iya b0tiin eded oxunda byin olunub (-
"o;io). lndiartna Ye azakna anlqlannr tapaq:
y' =l. 2r y'=0:+l-2r=O=x=f .
B u hatda arrna ve azaha a ra trqhn (-., i] " [i,.-L "r.I
'2 12
I2'
t'(x + 0
f (..)
Beblikb, (-.,]] ,"r,e .a" firnksiya artrr, [,*.) u'unr* .-,".
669., = G t,zol- r+100Halli Ye,ibn funhia r=-100 noqbsindan baSqp haryede byin
olunmrqdur Funksiyanm 6nmesini tapaq;t_l-.(x+100)-Jx
y' = Z'ti 'r r+100-2: -r+100
(r+roof 2Ji.(.r+100)2 2,[.(.r+roof '
u9
/'=O + -r+ 100 =0= r =100.
Bu halda eyin oblsta;a@kr kimi olur.
(-.o,- roo) u (- too,toolu [too'ro)
0,100 100 100,+o
t'{,) + 0
tG)
Beblikh, [00,r<) salrlmda frnks iya azalrr, (o,too] anhf,m& artr'
5?0. u=t.EcA IG) firnts iyas r (- -,ro) anhf,rnda tayin olurmugdur'
. 2x.2'-x2.2'ln2 iz- rlnzlY =----V;-' ' = 2, '
,=0, ,=?o2 nOqtobri funksiyam stfia gevion n6qtabrdir' Onda tryin
obhstl
1--,'l.-,fo,fr].[#,*)
Bebrikb tuohiya (-.o,ol* [#*) anlqhnnda "-* 1.,#]anhlm& arfir.
Qa$nalo:
Ay@*tlunXiloba, @ort v" @,tlns oa$lomt upn:571, y = xne-, (n>0, .x>0)
Cavab: 0<.rcr olduqda fimh iya artandr, n <x<ro olduqda azahndtr'
672. y=12 -lrr2 (n >0, r>o).
120
- @,0 0 0.2'l[22
1"2
2
-.lolrnl'
t'G 0 + 0
tG)
Cavab: -o<r <-l ve 0<r<1 oHuqda fmlsiya azahndtr'
. -l<r<0, l<r<+@ oHtlqda arhndrr'
673, y=3t-,3 .
Cavab: -o <r <-l Ya l<r<{o oHuqda frnk<iya azahnd['
- I < r < 1 olduqda albndtr.671. 2xv=--' . -
l+x'Carab: -o<r<-l ve l<r<l'@ oHuqda frnlsiya azahndr,
-1<r<l olduqda aftandfi.675, y=r+sinx -
Cavab: Funls iYa artandr.
g & QABARTQT IGIN isrlQAMUrL evil,uar@raslto. Qaboqh4 frqiin z*ri {tt fuet 7=7ft) tunfs iyas nm ayrbi
ayriyo gakilmig bxunsnm yuxarc nda qalG8, onda eyri gokilt toxumnm
a$ag6 Dda Slrsa eyri, gbanq ayri adhnr.' Sget aitrc^irltanan v =,fG) firnksiyasmm ikinci t'rtib tdnmesi
1'(x)> o (a < t< b) ohsa ayri 96k0! /'(r) < o oHuqda bs ayi gbanqdr'
Bu syrinin qqbenqhlr va 96k0kt0yti 09lin zaruri |aft adhPt. . .*. ayin, nd,qbs i nqtn kafr larr Oyrinin ktlarmtinin dayidiyi noqb
ayih n0qbs i adhnr. Ogar n nOqbsinda .f'Go)=o vs ya
j6;-yoxausa bu nOqta ayihe noqbs ilir. Bu noqtani kegdikda ftinci
tiil ora"lo oz iansini dsviise bu ayilm nOqes inin varl{r l&th kafi
gort olur.675. y =t+{i ayrbinin (-tol'(V) vs c(o,o) ndqtabrinde
qabanqkfnm btQarmtini mibyyan edin.
Hallt Ona golE kinc i brtib toEmeni tapaq:t
Y=1+ x'25
,,=1r-I. u. = -2 ,-1'392l9!
r3
,ru)=1,0.,{(- l,o) noqas inda ayri 96Htk eyridir.
12l
7y.(B)= -;.0 eyri qabonq ayridir.
v'(c)=o c noqtes i ayilma noqtts lfu.t
677. y= aa ai frnks iyas mn qabanqhq araftqlannr va ayilmandqtas ini tapm.
IIalli2t
Y'=1alll' lo -- lo I
' 3 v=;* '=t'3Gx>0 olduqda y'(r)> o ayri gOkii( r<0 olduqda y'(-r)<o ayri q4benq olur.M(0,0) noqt s iayilrne ntiqtasilir.
67t, y=ul+i) frnls iyas nrn qabanqlq aralqhnnr ve eyitnen6qt s ini hprn.
gerli ftinci brtib tiremeni apaq:
, zr , ^t+ x2 -zrz zb- r2l, = t*t., =,TlT_=G;rlr1>l olduqda y'G).0 eyri qabanq, Lrl< t olduqda y'(r)> o
ayridir. ,1,(- t,tn z) vo ,{2(LIn2) nOqE lari ayilme noqtabddir.679. y = xsit(hr) (r>o) funla Vasmrn qabanqhq arahlnr
noqbs ini tEpm.IIaIll Funls iyann ikinci taftib tdnmasini bpaq:
y'= sin(lnr)+ cos(tn xI
," = 1*.16,;- lrin(ln,)= 99$r--'in(hl) J' /. z\=
-c!c Irt + _ l_.r [ 4)
L +2kts <lnx<5A+2kE (k=0,r1]:2,..\ amhfmda ]'<o eyri qabsnq
ayidir. -3a+2*fi <lnx<L+2kn anhf,mda ),'<0 eyri gokitkdiir.
1+2kr 51+2k*
e4 <t <e 4 an [f,mda qabanq,
noqtes i
bo ayilme noqosilir.680. y=x'(v,g) ayrb inin qabanqhq btiqametini va ayilrne n6qbs ini
tapm.122
ayri 9ok{lk
ve eyilma
Beblikb, eyn1+)k* I
e4 I
izl)
CrI-11 *ztne4
s +2tt ( o,o,<x <e4 aahtmda 96l1lk tyridfu,l e4
t
Halll
.y'= r'(tn' + tI r"=r'[tr"r. rt'*1].
r>0 olduqda ,*i(tnr*t)'*1lro "rn
cetkuk oyrxfir' Umumil'1ctle, r>o' L' x)oldu[u iigun syri gok[,k aYridir'
681. J(x)=3x2 -x3 funksiyas"t,rr qabanqhq, gokikliik arahqlanm va
evilme niictelerini taptn.' Halll' Ba.rrfan ftnksiya ad5d oxunun boEn n6qtalarinde
diferensialhnandrr. Bu funksiyanm birinci ve ikinci tartib tirsmolerim
tapaq:
l'(i=a" -t*2 , /'(r)=5-6'=6( -r)'Aydmdt ki, (-oJ) arahpnda 7'(a)> o,(r;+-) arahEuda /(a)<o',=t
ndqtosinds isa /'(a)=o ohu. bu onu gogarir ki, barulan funksiyann
qotti 1- orl ar-alrE;ruda C6kiik, (\+rD) aralEtda qabanqdrr; ue (tZ)
n6qtesi o]'itnt noqtosidir.
682. 1()=i+rz firnlsiyasrnn qabanqhq, gdkilkliik arahqlannr ve
avilme nddelorini taPm.' IIallL Yeilmiq ftDksiya but[r edtd oxunda kasilrnozdir vo iki defa
6lifqsnsiallanandtr'. Bu firnksiyamn I va tr tertib tdremolorini tapaq
f(')=+=--'l'(-)=l-JI +."
t .,rFAydrndr ki- vr e(- "o;+-) nqiin /'G)>0. Demati' baxrlan finkstya
bii,tiim aded oxunda gitkiikdiil - Bu firnksiyanm ayilrna ndqtasi yoxdur'
Qabgmalot:
Ay{ulah lunksiyahnn qab@qhq, g ka ah anhqlotnrayilma nAqtalarhi tqn:
683' "= =o3 ^ 1oro1' ol +x'
Cavab: lrl> ft -de qabanqdr, i!'ft i'*"rtattattn& f'oki*dtu
;; = 1 a= eyilme ndqtesidir.,, J3
681 !=e-,2 Cavab: Funksiya gbanqdr.
685. .y=r+sigt.Cawb: 7Ztr,1Z* + l)r){e qabaqdlr, ((2h + t)t,(2k + 2)t) itrt rvaUannda
Cohildtu. x= k4k eZ eyilne noqtesidir.
s 9. QEYRI - MtrOyyONLirOORlN ACTLMAST
L lnptat qaydast. (f, SaHinde Oeln - moarymliHann a1tlrrsst).
Oger l) 7(r) ve g(r) fiEksiyslan q n6qtasinio etrann0a tayinolumug, kesilncz fimksiyalardnsa ve r -, a olduqda her iki fiuksiya sfirolufsil: lim /G)= tin g(r)=0.
r-+a ,-+o2) a n6qtesinin eEafindr (a ndqtxi mtstasua olmaqla) 1'(x) w g,(r)
tEreualori eyni amanda srfr dellse (: * a; .
3) Sonlu vo ya sonsuz
li," /!'l*+a g'lr)
varsa, onda
ri',44= mlQr'+a gF) :-+o g'(r.l
do[ndur.
II. Inpttsl qafiost. (9 S*ilda qeyri - mfiayy*tliklann ag t$t ).@
Ogar l) x -+ a olduqda /(r) ve g(r) somsudup gedirso:
E,r(,)=r-sG)=*bura&, a edad vc smsuzluq ola bilar.
2) o- lrtt istonilen ctraflda /'(r) ra g'(r) t6ronolsri hiuin r-ler ilgiinrrana vs eyni ""rnanb fz|)+ g'2(x)+O
ri.44 u.lQ=ri,"lQ3l x--+ag'lx) sonfu ve ya sonruz limiti vars4 66da tl.glr) r+a g'\x)
berabediyi doErudur.
Bagqa buAia qeyri - mileyyenlikler miixryyen gevirmslarin kOmeyi ils Iy3 3 gcklinde qeyn-mffeyyeuliklors gdhiln.
686 tim t$:.x limitini taprn.
r-.)o r- Srnr
Holll Bt I onft"O, qeyri - niieyyenlikdir. Odur ki, I lopital
qaydasmr tatbiq edek. 0nda alanq:
1--r--Ji-'-*- t-ory"5o I - cos x ,*-o (l - cos r)cos
2 r.. I +cosr -
= = l!trl ------.=- = Z, _,0 COCZ:
6fi. tu!( + - Ll rririoi op,o.,-+ox\tlc tgl. )Helll Bu (co -.o) geklinds qefri-miayyonlikdir. Odur kr, sadalagdirne
apardrqdan sora lopital q4ydasm tettiq edak:
li^t9,- fu = li^ w2 t ch2x
t+o ttlttgx ,-olntgr+ltgx+_i" Uach'x oos_,
= lie t+tg2x-(t+th'r)
''o ttotg, * -!r-Er, -+ilnch'r @s - x
^-7** ,Yo,,m
limitini apm.
Halll Owelcp -L = y avezlenmasini aparaq ve Lopital qaydasmrta'
tstbiq edek:500
lim /''- =500 ri, a=0.y-+@ eY y'+@ el
6E!). $m x" -i limitini taprn.I.,+0
Halll Bt 00 - gaklinde qeyri-muelyanlikrlir. Owslcs a' = r'n'diisunudaa istifide edek.
,o,r,.d5 rornr,15.r'I1w= lim e tht =€Ho '.o rh, =
t++O
- ,,^ t2 -?- ffotlo tC, I tgx . L,rr - 3
T--;-x t COBtr I CGo, I
r25
rim xta2xtn4 lu rtnrr=lrht=l t.rQ=sr-+o t-+o t = e''*
,--t r--+ -rE2r,n,=c x =e '' =ee '+o =l
6* )y*(r'-r) ri-iti"i t pln.
flalti Bt oo - f.Hinde qeyri - m[elyenlikdir. Ora gpta i =s"bud[strundan istiftd. edek.
e = $n( i - r) = ri-(r" h, - r) = r,l--.",So' - Irjo\ ./ r-+0\ )
rim-r' =1 olduEurdan t'.*(l' -r) = -r .tut.
I
69I. ti,.,r.t; [mitini upm.
Iblll Bu 1- $eklind. qeyri - miiayyanlikdir. z' =e'b' distrrrurdan
istifrda edsk:L
IinP rD.+ -r lw = er''ll'' '-e"l-' ="-t =;'
&2 w=ttrrlz- #i tiriti"i rpr.rrl-
EChL Bu l' pklindo qeyri - miayysnlikdi. t' =c"hu disErudanistifrda edsk:
** rn(:-r) '5----t-
[E E=.ti(2-t) -' "eT in'+ :*=er4 t =e t =e '=e".
693. ttu kstf" limitini aPra.
1
Ealll Bt 1' s.klindo qoyri - muayyenlikdir. u' =e"bu dtshrurdanistifade e&k.
I
. ri, I*4 L2..r-, - hY =:
r-;P:{-t-tgyu -""itt*t'c - "'\ocu =r''i
- *'u ==" 'i"-'*' ="-' .
4
126
I
*n H(o#i); ri.itioi ur.o.
IIa i bt o0 gsklinde qeyri - miisyyanlikdir. Yern r' = e'h'diishrundar istiftde edek oada
r | *(b+tlzra, -_ .1 ii--ln, o ,T.. _lL *"r-- 'o4-
fi- lro * l'=r,;-' -zx+l =e - 2r+l Zr+lr'-+.o\ - 2x+lJ
t,-,,,tEl:+o. t t+o2.r+1
=e zx+l = I
liE.r-. JB j- sttr
- cd-
=e 2x+l 2r+lI
12,+t12
6gs, n=fi,,(o'=-,tn,l7 t -toot tunr.,-+(br _ rhDJ
flalli Bu l' Fklind, qeyri - mneyyenlikdir. Melum z' = e'b'dii'stunndan istiftda edek:
. I . a'-tha
v = e'10t7 b'-x!I,b -.. bx -rbbIm --r--\ ,
++02+r -xh4l--e
_,i.{+', ?'o) ="}(',*'.4l(l"'-n'o)
flaIE Bt o-- pklhda qgyri-mleyymlikdir. Odur ki, bu
elemeirtar gevirrm aparnaqU I Soklinrls eeni-mrryyenliye g*tok:
-. /coo.r l\ -. rcosx-sin.t .. cosr - xsinr- cosx!"=lrltrt
--- l= llln-= lrm -----------
'-o(sinx x,l x-o rsinr x-+0 sinx+ xcc.t
-. rsin r -. sinr+xoosr 0 -=-[-E--=.-=-18--- =-=Ur-rosinr+tcos.E t_rocosr+cosr-xgl.n.r 2
_ .t697. *=Y^\l*'E -c [mirhi tafln.
r-+O X
69a ,= bl"ts,-1) ri.itioi av,o
LN
limiti
la
h b h
rb Itr b
b
Ealll ht I rUioa, qeyri - miieyyr ikdir- Burada z' = e"h"
dEgrundan istifrda edek.I u(r',)
.. e,'""'' -e .. ,-(t+r)ta(t+r)_y/=Ltrl-=ellm-- -"-,Ti I -iio x'z(t + r)
I
=r 15 -ln(l*I)-' u- -t-j=-:x-+O 2x+31 x"+0 Z*bx z
Burada riq1661r)= ro riq(1* ')] = he = t olduBu oearo ahnmrgdr'
- I ltmitinitaorn.ln(l + r).J'- C;c'+a
tatll Bar:drz 3 gaklirda qeyri-miieyyenliyi@
Iapital qaydasrnr atbiq edak:
gstirerok
l+r
699. n = y,,,@!t- 4 (a > o) limtini taEn.- t-+O x'
flelL Ba I ssuinae qsyri - miielyer ikdir' Arrhcrl olan-q Lopit'l
qaydasrm tatbiq edak. Owalcc y =(a + rf -in tdremosini aynca
hesatialaq: tx
I o*.
lfuir-r0
n(r.*,)-u(,t[+,'u(x + Jr + r'z !n(r + r)
(,.{;7)'
Odur ki alanq:
k y=*lo(o + r\+y' =In(q+')* r-Y ' "' a+r't = rfr"G *,) * -r-j = O. +[nf".,,1. ;]
zu.,r^- *(l -"w) fimitini tapm.
. .^4r4 Bu o' $ottiDde qeyri - miioyyenlildir. z, =g'hs d,i<n,n,ndanistifade edek:
2l;
1.6* l*r2,-
() \x ,t(?"-*1 -l-lior l:*"rf. I =[Ee \t - )=lime ,2x-r.ro\t ) x-+q r-r@
-* r z
= tip g l+r2 .tEt -"-i.r-a+6701. ly./latf hitini tafln.
Eelli ,' =",^, dtorrundan istifa& edek:ll
!e dr, d,r,ll
lim 0/.f =[me'he=time i =tine Fr-+0 r,o
_.r2 . I
= lim e e 'I' =e0 = Ir-J0I
zaz !*[""s]']F [mitini 4rntro\ x )
T.(a+ rf ltn(a+r)+ -I- l- r,1oo
r,=liD L a+rl _'io 2x
Ilalli Swallr;i misalda oldulu kimi t' = e'h" d[sturundan
edah sonra ise Lopital qaydasrm tatbiq edak:
-3'-arcsi! r, .Jt-*zI I t-dt irlt &c!rn r x2I- t
=r^"n,ql*'j"l" =.9'- .r..+or-r0\ x ) r-+u
' ,-,1-" ""' ' JIr ".,"' ''lnr I.{l-r. t^^-2,-- _ _,1!} argslrr _=tuqe 2r =e'a - --e'
;T:7l
'tsin t I
=eiy'--; =ei.
istifrdo
,, .E.(Y)'rmitinitan'n.flalL u" = e"b" dusnrundan istifada edek:
-!-"rw , *(r*"L.'s,ofttfi ,t b+ r'l
- -r+0
r@3r-lEll 'I I ii$x .. emx x)
'+ -=ln- u[ ----:--ti- { "-' l" = lim er' x - stJo 4,-+o\ r,,/ x--r0
-. t , ,ooa,-"in ' I r'- rcoBl-tint I Un Pt'19-"'llm- -..- : ^;.}b ,3 -"2,_* 3r.-
"x +a2x e,.r x" = eo
l -. sin, t
=,-;iT"; =t;I
704. i^( *",s\7 [mitiru taprn.r-ro\ .r ,/
Halli oi ="""" diisurrundan vo LoPital qaydasrndan istifade ederek'
alanq:
I aftEx
lime" x - c'+o
I--ti+x* I6-:'.'"r" -l_e2rfo rr(t+rl) =e2r*,0 lr =e l
I
,t y{ry)' rimitini taprn.
130
HalE Awelrtbro uylun olaraq holl edek:
I - ,4rsk-ln--.. _2Itm ?-
x-+0
_.: - ArtiE
1i^ A"k J'**2 | .. x-ll+/ Anlt
-Um--2r+o ,t -I -. -rArslt r- lla- -:
= eZt-o 3rt -" u
I.1;
I
llimitini taprn.
t , .1 r-bll+t) .. r-Lr(l+r)- llJa (l+lJ!
-
u6 .-:--
= "",-o r'{l+rl - e,-c t'll+!)
I
=e'-o2t(l+r) -e 2
I
m lj:--
I/. \:707. l'r.rr.l iarcros r l' limitini taDm.r+o\r )
HallL
,-1J1""..,)e,-ot \t )
zrE 1;,of gIlT rimitini taprn.x+0\ cru )
Hatll Owalkt msallarda oldu[u kimi s" = r'h' dfis[rn:trdar
Lopital qaydasurl"n istifrds etsok hell etsnekla, alflq:
cb -si,nxc,s-sls.cn6x
.4H)'=,si'H ="P'i#=-l r,.'ij"h'491l -l ra[fiJ"r**4-"') 1..
=" -'-u ' =, 2'-o\ x t ) =e 2' -e'l
13l
t-
zM l*lt
,* i"f"L
709. .. shcr-r0 sinDr
7IA .. cla-oosxrrE---.x-)Q t'
7II. ,. ltg4r -lzt(xtun--r--ro 3dn4r - l2sinr
712 .. tv3rtrf, -.. lqx
71J .. xrlet-lrlm ----=-x'+0 yt
7r4 * lq:t,-a12sin' x -l
4
Qohgnaldr:
Carab: 9.b
Cavab: l.
C,au : -2.
Carab: 1-3
Carab:
Cavab:
Carab:
Carab:
Canab:
132
_l3
I3
I6
715. [m r(e' + D -_2(r' - Dr-+0 rJ
716 -. I - cos 12LtIr ---:----------; .x-+0 I sktx'
717, arcsin 2: - 2arcsin x
Caub: 1.2
Carab: t.
Carab: l.
Caub:0.
Carab: lur.6
Carab: -2.
Caub:1.
,7linr-+0
718, ,. Arsh(slr\ - Anh(siox)[m--,-+o ,rrr - sin.r719' ri- { 1"ro;.
r-t+@ t'720, .. d - a6,
hm -: la>ul.x--r0 ,J
721, .. r' - rlrm-.r--rl lnr -.r + I
722 .. In(sinar)
r-r0 lr(sin 6x)
723. ,. In(cos c)tlmr--ro (cosDr)
724. .. cos(sin r) -cosrllm----..r-)O ,4
(;)'I6
725.Iim a (a>O. r>O),
x-+.t< edY- r2"4'olt
tim lnr.lnO - r) .r-+l-D
lim relnr (6 >0).rJ+0lim r'.
,!-++0
li-(aF)"".
u-full'.r-+o\ r,/
Carrab: 0.
Cavab: 0.
C-avab: 0.
C.avab: 0.
Cavab: l.
Cavab: ek .
Cavab: l.
Cavab: l.
gur"$' "Gz.
Cavab: f ,
2
Cavab: 1 .
Cavab: oogna -t;.
Cavab:
I
Cavab: e-i
726
727,
728.
729.
730.
731.
732
7i3.
734.
735,
736
7i7.
73E
h
um tr.+nl.r-++0
1.+f, r-tamauy.
.. (r 1 )i$l;-", {J.. (t r \llnt-__t.x--ro\hx r - l./
ti^ I -'" l o ro;.
x-)a x-aI
n-rE.)7,'rO\ x J
II 'l;rulo*'l' 1r-+01 e Itl
t ,cthx.. Il+e' ]lrrtl I _ |
x-+o( 2 ,J
h-!IE _.
x--r+o Qn r),
1e3
739.
740.
Cavab: €.
Cavab: 0.
133
,r,' f Fl"u('-").r-ro[Bo,
$ 10. TEYLORDtISTURU
to. Teylorun lo*nl teoiemi. Ogar: I) /(r) fimksiyasr ro n6qtosinin hrr
transr lr-*ol<" atrafinrla tayrn iunmt5sa; 2\ f@' il bu strafrla (a-t)
ter[b qpder tiirenslari vars4, 3) ro noqtssinda /$ varse onda
fu)= L'/g-,of +o(,-,oIt=0
(l)
dolndur. Burada t$)
,, = 1# (* =0.r,2....,n\.
X[$ti halda ro = o olduqda alanq:
/b)=,i+,'.0(,') (2')
Gkterilen .trafita (1) aynl4r yegaledir' (2) drsnruodan isa bel esas
aynl4 almr:
l. "' =t+ " +* r. -; -o(r'),
I ,-r .,h-1. * 0f,2,1.IL sinr=r-L+...+(-l) (2r_ Ut
m. "-,=r -{ * ..* 1-rYffi * o("'-');
rv (r+tf =t*.*t^9it)"* *
tu)*-.)t *o({);
v. rr(r +')=,-{ +. +(- rY-rL *sf,')
zo. Teylot dxruru. OFl,,t l) /G) funksivasr [a'a] parcasnda taln
oluonugsa; 2l fu) fiEksi)asrnrn b,tl parP$nda kasilmez
,'6y,,7b-r)61 tdremelori vars.u 3) a<x<b aral$nda /G)
firnkdyasmn ,(!"i sonlu t6ronesi rarsa onda aga[rdalohr dotsudur
/(,)= t' /(:](")G-af + a,(r) (o < x < b).
tt4
46.y_ tb)b *!G _ dl(r_ af (o<a < r)
(Qalq hildin Loqranj sakli).
-,tg = \ryl/l(t - qY, G -,Y (0. q . r)
(Qaltq haddin KoSi formast).
Gtstarilan q[r"v"atlcr daxil olmaqla a*.Sdalo firnksiplann x - inmiisbat q[wetlerino g6rs ay rgmr ya.an.
"r. ^n='i#; ,r -e@er 7@)p1=t
.
I/r//i Surct vo me;raci (t +r)-a vumra4h, venlan fiurksiyanr
71,1=r+(2,+z,r[+/)t'kimi gdst rak.
IV aynhgna g6re
(*,'f'=r-,'*o(')yazanq. Onda
/(r) = r * b, * z*'\*,rfr = 1 * Q, * 2r'lr - I * o(,')==t +zx+ zr2 -zxa +o!a),
/r\o;=-rs7a. .fi-r;;7 -{14'-; .t' -ys qeder.
flelL
lG)= J-.''' =(,- o.lF = [ - (r, - "[ =, -](r, -,')-i.11_r') t.l'1_iYl_,)
_ I ll 'l(2,_,_rl * 2 t2 -4.2
'./(r,_,rl -oF)=
=, - l(r, - r).
;(r, -,'f . *(r, - rl . 4')
i,G)='v!:1, -,' = [ - (r, -,'$ =, - ;t,- ri. iLilO' -,'t -l. fl - r)f1 - ,)_l\, tl, /6,-,,)*(,,)=
=, - l(,, - r)-;(,' -,'l - fi$. - "'l .,F)
l(r)= lt(x)- 126')=],' *,' *o(,').
74i. "zx-x2
. 15 -qadar.Eelli: I rynlug 96rc yaanq'.
J$\ = sb-'2 =, *Q, - r\*b'il t
*b' -tf *b,':'Y *6, -n'l *06,;
71'y=r*z'+*2 -1r' -:,4 -+,5 *o('5)
,nn lG)=GW*; 12-naqadsr.
HaAL IG) firnlsiyasrm clveriqli gekilda yazaq:
7(,) = ( +,)*(r - zr){ (r + z,)*IV aynfur tetbiq edek
[+r]o =1a166,* too'(Lm-t)t:
* o(t')=
= I + loor+ 50. 9912 + o(r'z),
[r - zr)-{ =1* so, * -4(49-l) 4r, +06,)=
= t +8or + go. 4 lx'z + 0(x'?),
(l + 2x)-s =, -r'r* - eo(ok t) + r' + o(r')=
= I - lzox + tzo. 5gx2 + o\xz).
AIdrFmu bu a)mhglan tarsf - tsrefa vumb, 12 -ru saxlayaq. Onda alanq:
7(,) = (r + rlo (r - zrf (r + zrf = r "
oor + lrsor + o[2 ).
I - V aynhslannt tatbiq etmrkld, llmitlai taptn:
zas. rrL-,,"].,.1)1.'--L \ '/JHatti Bwada t(t. i) =
+ - # . o(j)
"v"r's.a." isrifide ersak:
.*[, -,,'(, - t; = y{, -., (i - *L.,(i)] =
1/1 \Zta tn:l! -cWl.r+6r\r lHalll Ovwle misah alverigli gekilda yazaq:
,. I sinx_ roos xffi _ '
---,--)@r fS[tI
Burada sinx vs cos x fimksiyalanmn aynhgud^n istifrda etsak:
.-t'.d,)-{'-*.-ot'))lim . ,=
r'(r + o(-rr)
-lHalli m x = x -,L+ 0(rr -- r + o(E3 ) aynbgrndar istifrdo etsek:
,8,(: *)=.{i ,*d=.y.(: i)=,
Qolqnalot:Gddafibn qtwatl dcil dnaqla ogapfuh funhsiyotofln r_in
,nL*il qilwailait agdn aynhyu yun:qlan + x (o>o), ,2-na qedcr.
cavab: a+_*= _ffi*arr1.
fr , ,o.nq"a"r. cavab, r_|_{_!_al,rnccx, r6-yaqsder. cur"b,*_*-*_nr,sin(sinx) , x3_na qader.
Canab, ,_4 *r(r3).
=,g-('-'.;-'t(i)=i
= timl-r@
ru2. n(!--L\,,or-o[r sinrJ '
751.
'-*.'(")-'.* -o(") ,
3
117
rgx, ri + qader.
lo"h" , ,6a qadar.x
cu"at, ,*{*]{*,t 'r'
c.,"b, - + - r8o4
- ;l. nr,.
{bvab: -}
I - V oyn$ltnn tdbiq ana*la' hrtd*i tapn:
-'2 ca,ab: -l'-. **-"-n 12
lrln
----.r-+O ,4
-. e'sinr-r(I. xl Cavab: l.ru[ -=--_i- 3r-r0 rJ
751
759.
757.
75&
19
90
It
I1
S II. FUNXSIYANIN EKSTREMT'JMU.
rum<slvimi{5n sowK ve oN IdqtK QtYMorLoRi
lo. Ekstremurmtn vaf lt& 91ln zerui iart'Ogsr 1@) funksiyasr xo n6qtesinin istanilen 0 < lr - r'ol< d otrafinda
ttayin olunmugrlursa ve /(r)< /Go) w ya t(x)> f(l,o) prtlari 6'denirsa'
lm y2(./r+1 -Jx'l -2.1x).
,ri- (E[tJ-1tr=) c"*u']
* l1,,.-,r*r1"i rr;l ca'rau I,-+-l\ 1.) lt o'-ol"'-2 (oro).s-ro r'
Cavab: ln2 a.
76A .. sidsinr-r3ill7um .'r-r0 x'
761. .ritr r.. l - tccir,[m -------------:- .
rr0 r!m d(E9rr-+O tJ
Cavab:
Cavab:
Cavab:
751
756
762
onda ro ndqtssi /(x) frrnksiyasrmn uylun olaraq rnaksimumu (minimumu)
olur. Ogar ro ndqtesi *$rernum n@sidiss, onda 7'ft0)= o olrnahdr.
20. Ebtrenruntwt varh$t frg i W tert.Btrinct qalde. Ogor l) /(r) fiuksiyasr rb n6qtasinin l, - Al . d
,tmfilda t yin olunub ve kesilmazdirse, 7(ro)= o va ya tireme yordursa;
2) lld -n o<[-rol<o atrafinda sonlu toromasi van4 3) 7ft) funksiyasr
ro ndqtssinin sa! ve sol terafin& 6z iSansini saxlayusa, oda fimksiya
aga$dalo kimi xaraltcriza olunur:
Tannonin ismasi N*lo
IlmIv
++
+
+
Ekstremum yoxdurMaksimumMinimmEkslrremum yoxdur
Ibinci qa1fu. Ogar 7(r) firnlsiyasmm rs noqtesirde ikitrci tcrtib
/'(r) toremesi va$a vc
1'(a)=o vs 1'Q)*ogertleri 0denirse, bu noqtede funksianrn ekstsemumu var ra /'(16)<0oldu@ firnksiyamn nraksimumu, .f'(ro) > 0 olduqda milimumu var'
(19'ttnca qayda. Og.r l)/(r) funksiyasrnm l"-roi.d *rafnda
/(r, ,.r('{\') 1€ ro ndqtosindc /o)(t) varsq, 2)
f(,\ro)=o (r=r,2,...,n-r) lt1o1+o prtleri odenilirse, bu halda: 1)
z-crl sdad olduqda 6 n6qtesinda /(r) funksi1almn ekstreraumu var vl/(')(r0) < o olarsa makrimum, ,rfr)(ro), o olarsa minimum ohn.
7dl.. y=2y2 - va fud<siyasrma ekstremumunu tapm.
flatli Frmksiya (- *,r-) aratlode t yin olunub ve kasilmozdir'
Stasi@ar noqteni tapaqy' =4x-4i + y'=O=4r- 4r3 =O:)
= +r( - r'z)= o = tr(r - r[r + r)=o ==4+0,r=0,x=l,r=-1.
stasioar n6qtelardir. Bu halda teyin oblasn (- .,-rlw [- t,o]w p,tlw [r.+-).
- @;-1 - l;0 0 0J I l;+ol'(,) + 0 0 + 0
tb
r=+l noqtalori furksiyanm maksimum ndqteloridit. r=0 n6qtesinde
fiulsilanu minimumu var. max /(r)= Lmin /G)= o
764. y = ys-r fidtsilrasmm ekstremumunu taPm.
flalll Flrn*siya (- o,ro) arahlrnda teyin olunmugdur. Stasionar
aoqtaleri rapaq:
y'=er -*x t y'=o 's a*(1-r)=o+at*oodur ki, r=t nttqtasi stasionar n6qtadir.
-c.ll I l;+co
ik) + 0
/(,)
x = I n6qtosinds funksiyanm maksimumu var:
maxflx)=7(r)=ar = l '
Qah.pul*:Aga$.loh lunhsiyalann veilm$ paryala a an b|yfr* va an Wh
qlym bini toptn:765 1(r)= 12 - ar + e; [-3;10]. Cavab;2 vs66.
767. f3l=**:; [o.orloo]. cavab: 2ve100,01.
76E .. ( rz ,,) _- , . Cavab: Oyel./(r)=l t + r* 1+... + :- l€-r ;(0,+@)."'l 2 'r)
769. f(rl=a,' *",r; (--,+-). cnu*, _Efti *_{,0067 ve t.
2
77A y=,1;*; l-t,tl. Cavab: I vs 3.
771. t=12 _**zl; [-ro,ro]. Cavab: 0 vs 132.
AEapdah fu nksiyalann ehstrenumlonm tqm:772 y=r3 -612 *g*-4. Cavab: ,mir(3)=-4, /mrx()=0.
140
773.
774.
775
776
IY=t+-.xx2 -3r+2y=-"-.x'+2x+l
Y = Jito,ca,tab: ytu(e)=-z-
s12. XARAKTERISTIK NoQTOLORO NOZORON rUNxStyAQRATIIdNiN QI,IRULMASI
Funlsiyau a@iq etnok ve qrafikini qurmaq figln a$agda gdstarilonqayda ila borekst eunak lazundrr.
l. Funksi;6111s tayin oblasbru tapma\ periodikliyini G oxu ilekesigmo noqtclsrini, sabitlik amlqlaflDt frrnksifafln qrafikininsimmetrikliyini, kasilrne nOqtslerini vs kesilmo aralqlann rdayj€netnek larundr.
2. Asfunptrtlann varhFfl m[oy]€n etm.li.3. Funksiyanm monotmluq araJqlam vc ekstemum nd{alarini
tapmdl.{, FunksiFnm <ptnnqtErrt, 96k[ldtytn[ rc syilrne ndqesiai
tapmalL5. Funksiyaun qrafikinr qunnat.
77t l@=d;f tunksiyaru tadqiq edin.
Ealll l.Teytn oblasor-r tapaq. Funksiya r-in x=-1 qiymatindenbagqa hsr yerde teyin olunmugdur. Belolikle flrnksiya (--,r) ve (t,+-)futerrallannda tayin olunmugdur.
2. Simmefiya markozini ve simmetriya oxlanm tapaq. Bunq gOE)
71- r)=7(r) ve r( ")= -JG)berabertiklerini yoxlayaq. Bunlann heg biri 6demrt. Funlsiyanrnqrafikinin na simmetriya m*kazi, no do sinmetriya oxtan vadu.
t4l
c.!ab: ),D.x(o) =o, r* (i) = "iCavab: y-o1t; = e-l = 0,36t.Cavab: y*1-t;= -2, y^6(t)- z.
c","b,r*(1)=-;
, r{+o) = o serhad maksimumu.
v
vl+x2'
3. Kesrin surat ve ma:rreci kesilmez firnksiya oldupundan, fimkslya
.r = -1 n6qtosindsn baSqa her yerda kosilmezdir.4. Funksiyaun qrafikinin asimptolanm tapaq.
a) Maxreci srfra barabJr etsnekle asimptotu tapmq.
2(x+tf=o=r=-tr= -l funksiyanm Saquli asimptotudur.
b) Ufiioi asimptohr tim /(x)- i tapaq.- .t-+lco
t-+l'- I-ris 2('r + lfBu oru gdstsrir ki, iifiqi asimptot yoxdur.
v) Maili asimptotu y = kr + D $eklinda a{araq:
e- tio' 'rG)= lim - '3.;=1,rrio r x--rt62F(r+l)' z
, =,rm L/G)- hr=,,-*{r+ j "l = i,g t# = -,
,=*lr-1 funksiva eyrisinin maili asimptotu otur',25. Funksiyanm artna, azalna arahqlarmr ve ekstremumunu tapaq'
Buna g6re birinci tortib t6remoni tapaq:
,,r ., 3r'(1, l)':2(tt-0,' ={G!)J.\,)=_A*ii- = zt, _ rt,
l) Bdhran n6qtelerini taPaq:
r =0,r = l ndqtalsri b6hmnnoqtolcridir,2) Funksiyamn torsmesini sonsuzlula gevlrtn noqte x = -l ndqtosidir'
BeloLkle r = -l n0qtesi tsyir oblastna daxil <teyil
a) (- "o,-ll; b) [- l;-D; v) [-r;o]; [o;+*) arahqlan fi.rntsivarun varhq
oblastlandu. [Ier bir aralqda t6roma <iz igarssini saxlayr:
-o,-3 -3 -3,-l - 1,0 0 0,+o
f'{') + 0 t vitrolunmaub
+ 0
f(,) -t"
Aydrn gdriiriir ki, firnlisiya uy[un olara1afti, azaltr, orfi, arlur.
r= -3 noqtesinde birinoi tortib torame 6z igaresini musbstden menfiye
dsyigdiyinden hamin ndqteda fuoksiya:un maksimumu var,
142
^*yQ)=11-t1=-(r = -l,x = 0 n6qtelerinde birinci tortib tdmmo i5ansid
deyigmsdiyinden hamin n@lerda ftnksiyanm nralaimum w minimumu
voxdur.- Indi fimksiyafln qabanqhSm, gdkukliiy0nti ve eyilrne noqtesioi
tapaq. Ooa gtim ikinci tertb titrcmeni hesablayaq:
f,(*)= 1'--(r + l)a
ikinci nov bohran n@slerini taPaq:
3x
1';ry=o='=oBu x=o bobran ndqtesi ilc funksiraorn varfuq oblast (-4,-t) ve
(- 1,.,--) a$agdalo kimi olur:(- -,_r), (_ LoI (0,+@)
r=o ndqtosind, frmksiyann ikincideyi$iyinden bu noqtc ayilme nciqtasidir.
tofib toEmrsi 0z i$arasini
v
141
-o,-'l - 1,0 0 0, +or,G) 0 +
7G)
t) =:r+
s
77g. JGI=
= funlsirasrm tadqiq edin.'
''lx' + I
1. Funtsiya r - in bfrtiin qiymstlerinds, ycni (- -;+-) araJrts@ tavinolunnu$ur ve kesilmczdir. x-+2 olduqda fiuksiya rniisbe! x<2olduda ise firnksiya meofidir. r = 2 firnlsilasrnrn sl&rdu'.
Z. ,!ilG)=tt oldulunthn y = t funksiyaorn qrafikinin asimEob&r.
3. t'G)=#+ 1(,)=o=+ ,=-JJE'* tI
,=-] Utt - oout rUir.
I,2I-1 I
--.+co2'
t'(' 0 +t{, -"-
(--.-il-[-i,.,
r = - ] noCosinde trnksiyann minimumu var.
,./(,)={_;)= #= +=*{; *'
Ina nrntsifanrn qabanql'Ernr, fokiikliiyiirti ve eyilns ndqtasinitapaq. Bumn flgun ikinci tortib toEmoni hesablayaq:
7Q\=1t*z,lp'*rl1
r.Gl = zb, *rli -1G, . rli . zx. (r + z) = -!-. M =' ' J['.,1 J('' .,)r2r2 +2-3x-6x2 4x2 +3x-2
-t - J+t- rr =-
8
_1ffi_=ffi/'(r) = o =a -(4r2 + t, - z)= o,
-txJs+A -t+Jat -3 + J41'=--'-=8'I'=--
noqtaleri ikinci n6v bOhran noqtebridir.
-3-,tql-@,-
4-J4t8
-3-l4t -3+Jil8 8
-3+J+r8
-3 + J4tt
f'(,) 0 + 0/(,)
, = -'
;t ," , = -' *,fi ollme n6qtelsridir.
?so.
^n=+ fimksiyasmr tadqiq edia.
l. Funlsiya r=o n6qt siDden bagqa br yerde teyinol,nmugdr. Yeni (-o,o)w(q + -).
2. Oxlara vo markez: nezaran simmetriyau yoxlayaq:
ft+{-t
sinmetriyasr pxdur.3. Funksiyamn qrafikinin asimptotlanu tapaq:
-ll .r=0 n<iqtesi fioksiyanm kasilne noqtssitir. r=o firksiyaSrafikinh ga{uli asimtrdrtrutr totrlildir. BeloliUe y oror gaquli
Dofrudan d4
t^ /(t)= [m 1=+-, lim f(r)=I_+ lU I++O l, t+{_ ' '
b) Ofrqi asimptour tapaq:
,;- {*-, f,- {=0.x+{D X r.+--@ .t
Utqi asimptOun tsnhyr y = o - dr. Deme[ ox oxu tffiqi asimptrnrdur.
IIE - = --o,-r4 X
145
v) Maili asimptotu tapaq:e
l,= t,m 1= tiro { =.-' r+'_ .r r_+o x'
.t -, {o olduqda ril asimptot I'oxdur'e'
/..= tim -L= rirn4=o' t__t x '-- 1-
Bubaldadamailiasimptotyoxdur.Belcliklofunksiyaeyrisininmailiasfumtdu voxdur.--l-li-*ftiy , monotonluq intervallanu tapaq:
' e'rJ - e'(x - l)
f'\rl=--- r- =-f-1'$)=O a e' +o"r -1 = 0='=l
[email protected][ (- -,r]., [+-) kimi olur'
-o,l l, +*f'\x 0 +
-fG\
r = I n6qteslrdo funksiyanrn minimumu v'u'm-7G)= /(t)='
5. Funlsiyantr qabanq tfl, g0kiikliiyunU vo eyilme ndqtesini tapag:
1'67=F'-z'-+zb.r
/(r)-o berabarliyinden 2 - ci nov bohmn nd$slorini tapaq: e' *0
xz -2x-2=O) Y=l1-l
Demali r - in heqiqi qiymeti vs firoksiyanm eyilme ni@si yoxdur'
r=o ndqtesinde ftnksiyaeyilmo nti,qtosi dcyil
teyin olunmamlgdr. Odur ki, r = 0 niiqtesi
t4
- "o,00 0,+o
f(,1 +
,r(,
1
781, r= x; firnksivasmr tedrQig edin.
L Verihnig firnksiya y=li="i"' kimi g6stsrilo biler. Aydmdrr ki'r > 0 qiym+lerinds bu firnksila tayin olunub vo kesilmszdir.
/t I
2. /(-r) = (-r)(-'l olduEundan ftnksiya tek ve ctir deyil. Odur ki, ns
msrkoza, no de koordinat oxl,anna gdra simmerikdir'lnr
3 ,$r=,{i" =t v=l fimksiyamn asinptoufu'
4. Bohran noqalarim apoq:
1 , l- r-luy=r. =1.,.,y=16 rtL=' ; ,
,jt'
..,__if,_ l'\_Ir0 0<r<e olduqda, -^ [' ,, J-l<0 ecx<- olduqJa
Buradan ahnrr ki, 0<r<e arahlroda fiuksiya arur, e<x<laaralSmda funksiya azaltr, r = e -da funksiyanrn maksimumu var.
782. y -"'t fiuksiyasrm tadqiq edin.l+r'
l. r=*t noqtslarinden belg funksiya her yerde tayin olunmugdur.
Bu noqtelsdo fidsiya kesilir. Bela ki,v(- r -o)= o v(-r+o)=+"o;v(1-q;=r"o; v(l + o)= o
(- o,-t)w (-t,t)v (t.rc)Funksiyarun qrafiki q, oxura nezersn simmetrikdir.
,=,(,-5)=,i(, 5)
147
n ,IG\-- I(")="i
2. r=+1 noqtalerinda fruksiyann gaquli asimptotlan vardr.3. Funksiyann tdrsmesini rapaq:
llI
= 2*3"t-,2 ! "'- *O- x')'(l + x')'
y' = o + r = o, r = +J5 fiuksiyanm bOhran ndqtaleridir.
r = -JJ, r = or6qtelorinde funksinnrn minimumu var. r = J, ndqasimaksimum n6qbsidir:
na 1 Q) = 7Q\ = e, .*/(,)= /(JT)= +l0e 3
4. Funlsiyamn 2 -ci n6v bohran niiqobrini tapaq. Bunun ligiin 2 - ciartib t6rsmeni hesablayaq:
.,, -.-# . r .x6b-'2Y *'2i- ,2h*'2 *t'a - *61y=Le.-;7@Buradan goriinur h, I'1. r olduqda y' , o.(,Ji), (J1,l-)
"e
l4E
Q * ,2)2
- -;-Jl -J' -,11;- -1 -llo 0 0;t I r;.ll Ji Ji;+-
+ 0 + -t 00
(--,-J5),(-J3,-r) araqnn"aa tu*siya ctt oldulundan bu arakqlannher birinde bir ayilma noqt4si vadr.
Qalrywlot:
ASagfu wilnfrS fu n*tiy dmn qr6*h i qu rrnah:783. y=3x- x2
Cayab: Qrafik koordinat bagbn$crn^ n''zeren simmetrikdir. Funksiyanasrfrhl r=o, r=+JI.+L73 yon0) = 2, yuhl..l)=-2, Oyitme ndqtslari r:4y:0.781 , ,4Y=l+t_ --.'2
Cavab: OY orutra Dozarcn simmetrikdir, Funlaiyanrn srfrr
,=+fiTo+Lss. )',*(11) = l;, v-r(o) = l Oyitne ndqtoleri
,=tJo+OSS. r=t1.J3 lt7t5. y=1r+t\r-2)2.
Cawb: AQ,2) noqtesins nazeron simmetnkdi. Fnnkciyarm sx&lr=-1, r=2./nio(2)=0 ym!x(o) = 4.€!rilme noqtolari r=1, y=2.
to
III FOSILQEYRI-MTIOYYAN INTEQRAL
Sl. sADo QEYRi-rr0evvex inTEQRALLAR
1. Qeyri-m,iiayyen inreqrat haqEndo' Oger /(r) fiu*siyasr (o"a)
arahgrnf ayrn olunmuq kasilmez funksiyadtsa ve F'(r) = /(r) Scrti
odJrr.a, ooau rG) ftrnksiyasr /(r) funksiyasrmn ibtidai funksiyasr
adlaff.
-i(r) *. -y" /(r) firnksiyasrmn qe5i-mfrayyen inteqmh deyilir va
1 1$\* -'r(x) + c (c = eonsr)
kimi vazllr.i. Qeyn+i)ryyan inteqraln asas rassalari'
4 al1|Qtul=r(,\x;b) Jao(xltr +(') + c ;
i !A/(,W= A!IG)dx (A=const, A*o);
d ILrG) * gG*,, = I tGW * I sGW.3. kda inteqrallar cadvah
.-,+ll. l/,dt=! -+c (n + -tl;
2, i-la=tlrl+c (r +o),
- I foctgr + c7. l--8= <
'l+ x' l- arcclgx + c
r l--l--^=i'li*I."t farcsinr + c
5. r-+e=] ;'jt_r, [-arccosx+c
r 1pa,=r"1,* 6'trln";rlx' t 1
1. !d*=fi+" (a>O a+r); !e'&=ex +c;
t. Jsin rdr = -cosr + c;
9. Icos r& = sin.x + c;
10. !!=-agx+ c.sur' r
150
il. l!=Ex+c;cos -.r
12. I shxdt= cla. + c;13. lchx&=sb+c;A. l!- = -s1tt r.'' sh"x
$. !{=w+c.ch'x4. Asar inteqrallanw melodlan.a) Yeni dayigenin daxil edilrusi. Oger
j 7g)* --n(:) + c
olusr I fQpu =F(u)+ c, bunda e =fix) oW.b) Ayrma mctodu Oger 7(r) = 7,(r)+ /r(r) olarsa, onda
ITGW=1fi(XW+ItZGWv) Doyisonin avez olu nasr, Oger 7(x) funksiyasr kesilmoz
furksiyadrsa, r=p1r) fiurksiyasl ve otuD r'=9'(r) tOromesi kcsilmaz
funksivalardrsa- oada
ttGY'1Aeilh'Q\r,bsraborliyi doSrudur.
q) Hissa-hissa inteqrallama metodu. Oger u ve v her he.sl
diferensralhnan funlsivalardus4 onda
ldv=uv -ldudo$rudu.
kda inteqrallar cadvelinden isfifde lalbiq et eHa a{ofuhinteqrallann halli:
756 ta:\ta/r
IIallLI t r) i I " 1
fldr=il "z ,*-llpc=1,1at+1r iai=lrl +2,!i+c.' 'lt '1. /
787. t-!-.' '12 -5xtallL
tffi= -ita - r,) ialz-s,1=-i 9:ii- * " ='? J; s' * "'
2
7t& I e-.'2+1t'
EallL. & l. & l. & I I rt;3?=itT 1=lt r *,=l'T_*"oTr*"=3 - |.,il;J
.' Vr Yr
r .l:,=
J:6&cts f7 + c.
w' P4-'t + srDr
.EaIIi Sade gevirmo aparaq:.itu&r.&'
r * "r",
= J. ---7;-_] = rl -;7; ;\ =I +cos[-: - rJ - "*-l? - tJ
o(z-:\. 14 2l=rffi="no. tt'-*.Zalli Difereosiahn komayindon isti$de otssk, alan4:
, t r, dh')ry-a=7tffi.
Burada r' = 1 kimi avalems aparaq:
-tii=i T,4i*1*=*n"5AM.r+.
xr,tx'+lflalJi Maxrecda sade gevirm apararaq alarq:
+ c.
z t'\-- l+c.4 2)
= -"],. { - rl * " = +.,[,. (:fi..= +. [, +
t52
lr I a=l- = A= -l: =l, | '.h*,2
.&tb.!- 6=r-;51a=-J
\,,
7e2. t-!-' ./r(r+ l)
flalli Me,,rrac,b sade fcvire apa:raq:
. & , & *d16LrmE=rE.F=,'ffi*,=lr;=4=4#=
sitr.r .cos.r796 l-=+:+-.=h.
'o) ;mz * + b2 w2 tfleilL ifrr4ra}aln iBd. nunrd. miie'
sin.r.cqs r IA=#tJa2 sit2 r + b2
"ns2 x +, -b')
tln<ta irtcqraft agldakr kimi yaanaq olar:
ld
r ---1= z r"f + Jr2 * rl+. -- zhl6 +
'G + rl+..
7%.1 &
flarli irteCralalh fi:nlsiya uzerinda sa& geviroa aparsaq, yazanq:
, & , ,'& , ok'l ,b, =A=, -! =I , -=l "2,
*r=l€f,= t t .tt +r
=@ctgl+c=uctgax +c.
7s4. t-4:.' Jeb +l
flaUi lnqralall firaksip fizerinda sade gevirme aparsaq, yazafll:
r -4- = t --4: = t4: = t& =L' = I =' ,17 *, ' ; ,,le-zx *1 ' ,tr-2x ,1 ' JL-,P *, ' '
=tft,=-"l.Jill=-+-'.7ss. t#:lflId,.l/sl[r - cos.Y
Ealll (sinr + wx!to =d(sior - cos r) olduErm&n inteqralr a5a$dahgekil& pzanq:
. sin.r+coox . ,, . I
f ;-==:a = tlsmr - cos rf id(shr - cosr) = pinr- cer =tl={/siux - cG r
l2= !r-i at =li *, = l{rior-o*rE
* ".
cn gwirme apar4q:
sin2 r + D2 cos 2 .r
a2 sir2 r + b2 crts2 x
6*Vyb' *'' * u' "^' fiaQ' "ioz "
* b2 *2 )--
=h2"-2 r*b2*"2 I I 1 -T --. -,- _* ,=,1=
z@ _F)t, zdt=11i\fi+c=
= "=]-- {,' "it t * t' *'\ * "'a'-b'
nt. 1L.slD r
.[IarL lnteqralaltr ifadcnin ilzarinde/elementar gevirmelar aparaq:
& & ol'8 r)
"jrt-=;I.* ' =- r
z "z '9,Onda inteqah ap$dakr kimi pznaq olar:
,ft!=i.:=a= !4=nt4*"=' r 1"2 I 't,81
zee. !\$ tlon.
-ttsil+c
llattt irteqralaln kcsrin surot ve moxrocini x2-na b6lek.
in[eqraldt ihde ugiio alanq:Onda
Burada r+l=r avazlomesinden istihds etsok,r. dt r lr_JZl t . lrr_..f:r*rlt v -= r7rl7;51
* "
= ;6^17;72'. ri. "
799.
IIam lrxqralultt ifato uzerinde mfleyyen gevirmelar aparaq:
t54
7-2
Jl * r*2
Onda verilen inteqrah aqapdakr kimi ya.znaq olar:
^ l*1
--r='lrlt2 +il+*"t21+".n+2 | I
an. t-::g:.'Jr.,"*/[*,fIIallL In4ralaltr fada [zorinds mf,a1y*r gwirms aparaq:
rr z+2\
-1- l"T In+2 [ )
at-;;jh*l' z IIt)
=r(,.@_
= (r. ",[;7)-;a1r. *[-7iOnda verilsn irteqrah a5a$dakr kimi yaznaq olar:
t
(,. d17)-;a1r -,'!i7) = l'.'.[ -7 = {
=
I= 1
lia =2.!i + "
= z\F * 57; * "
Ayrrma wulundon istifado edarab inteqallan hesablayaL
srr. J{ - r}odr.talli Buzdg. r = I - (l - r) kini a}Ilsa4 verilen inteqrah yazanq:
155
---:==::.--- ----_:-/, *,' . /(r .
"f ri1 "
rr1 * ,2 '/t
+'2
I
| -r7 I
=trt =,i= ht# = *+.il;71. " =
,[r -0-,X-,]0a=J(r-']oa-l( -,] |a' = -1(r -,]oa(r -,) *
+1(r-xlra(r-r)=-(r {}t *(l -I)'1*"
uztl.f,rrri ldeqralalu kesrin tzsrinde mueyyan gevirmolar
,1 ,r *31 -l' ,r *31 27 (x+!'lxz -lx*!+r 3+, 3+x 3+r
=*2 -ir*g-.27!+t
3+x 3+x
Bu hdda verilon idearah aga$dafr kimi yazrraq olar:
!(x2 - t, + o - A-la = l -1"' * s- - 27tnp + t1 + c'
ultL'{/l-3rEallt r -i tnct*xs uy[un gakilda yazaq:
,=Jir,., - D=ll,-0 - 3,)=;- l( -3,)
Ondaverilen inteqral aga$dalcr kimi olar:Itl
Jr<,-r,ir* - |l<r -r,fi6 -r,trt=-|lo - r,iia{r -r,)*
"?t' t n-irtl I fl-3r)3* |to -
r+a{r - rn=-;'!l-ir: + i' t' -i"" *
" =
33252
=-1.o-rrlJ *1.0- rrlJ *"=-ll-2'(l-3r)J *".6'- -'' 15' l0
ut t.dx'r'+x-2
Bunuverilsn irteqraldl nazaro alaq:
\ e -!t e =1u1,-ti-lu.r+zl= lhll:]l*".3t r-l 3Jr+2 3-t' I 3 r ' 3 lx+21
sor IG;all.46+br
IIaIli Inteqralaltr kasrin iizarindo bazi qenirmsler aparaq:
r I , l' I l(,+o)-(,*o)]2G;fl1;S =
lG-,X,.6) I =GIELG."X.,+) I =
= r I r _ r 1'?- r I r - ? - , l=@-ofLx+b x+a) (a-tfil(x+tf Q+a\x+b) ('r"Y)
=irlir.(.:,.-;(* *l]=_ r I r ^ r -z1r _r1l- b-*lG*oY' G*d2' a-b\x+o t+b)l
Bu gevirme asasrnda verilen irteqral aga$datr kimi hesablalr:
dr{;r. do!.ds'l* *b=
= (,-,r l*. *]. #t"'+a{-rnl'r+Dl)+c=
Ealli lnqralaJ/'r ifadenin uzorinda milryyen qpvirmelar aparaq:
#=#4 ,4h=";t:++.t=
=4@-U=et-[rde' l+e' et l+e'Bunlan verilsn ideqalda nozen alsaq:
t P - t4* =r,1,'l - r,fr * el., = "l#l. ".157
r I l I'l 2 ,lx+al= -
C -8 L, -,- -,-al -
1,:;pl, * al. "eodre'l+e'
&7. lchzr&..EaIIi Malum d0sturdfir istifade etsek ya2anq:
-. ehlr+ Ich' x =
-2
Bu ifidani verilgn intaqralda nezere alsaq
)! t"nz', ia = !"hlx + )r + c .
EIN lchtclSxdx.EaJd, Hperbolik funksiyalann hasillsrini cema getirms dilstunmdan
istifade edek:
"1n"16r=!(chtr+ ch2r\
2
Axmcr ifrdani verilan intoqralda nazep. alsaq:
ltkhlx+ "nzxE =l- sh4x+ ! sh2r, c .2"84
IMtn xzlamalar d*il etmakla inteqrallan taptn:
eoe. yi(r-s,zlo *.Ealll Orrolca inteqralalt ifadanin itzarinde mtelyen g€r'irme
aPariq:
"(r -s,')'a=-fi,"'( - i,')oa(r -s,') (l)
Sonra l-5.t2=r evazlamesini aparaq, Axrmcl berabarlikdan x'1=!alanq Brmlan ( l) beraborliyindo nezers alaq:
- | .l-t .lodt=- I (,ro -rrr!,l0 5 50'Buolarn osasmda verilan intrqral aga$dakr kimi yaznaq olar:
- I 1[ro -,r r )r, - -]|.{t- -,"'1 * "
- -f [G4 - t - "']' I. "l0'r ' t-' 50( il 12 ) ,ol il 12 )
A loos5 x 16iia..HaIIL intnqalala.iEda 0zorhda sade gevirma apa^raq:
*"!"...6 1d=-sar.Js, lcosx&=(-r-2rf . J"ioi alrio rl =r ( ! 1 r)=[ - zrir,2 r*rlonr).*i,a(,rr)=lr,oz,- 25io-i, *r;17 t]rft,or,
Axrucr ifadani venler hteqralda mzs.e alsaq:
l5t
(L 1 9) 3 ? rr
Il,int, - 2,i,, : . +,i"i, laGt,)= i"i"i,- 1,i,2, * i,in 2 r + c.
t.Jatr. tllltgY34r.'l+cos'rEoll.L t+oos2 x=r evadamesini aperaq. Bu ifadeni diftrenslallasaq,
alanq:_ 2smxogtx& =d, coszx=t -l.
Bunlan verilan imeqralda nezeo alsaq:
i'? = -i'lL - l), = - lr -hr,D. " =
= -](*"*'r-n[ **r"[*"
s2 !-L -.el +"
IIaAL lnteqalah ifadoni bir qedar ba99 gshlde yazaq:
'* = .*-r=i+"p2 *e' .2ll+et I l+e2-r.-l
e-i = r ovazlomesini aparaq. Bunu diferensiallasaq e '' * = -^ oldullrnu
alanq. Bunlan inteqralda nszeo qlaq:
,i lt. = -rt -t- a, = -zrh - !-h ='t - | -'l+l '\ l+r,I+-I
= -z( - ur1 + rl)+ c = r[,-, - ri,., ri] *. = -' - z"-tr* zrn6 + "ii + ".
x=ocit, x=olgt, x=asinzt ua s. evazlamalar duil etnakle aygrdah
inteqrallan tapw.
EI3. !, & ,r, .
h-,,Y2HailL )=*n1 evszlemasini tstbiq €dok. Onda & =cosrdr' Bu halda
inteqral aqalrtlakr kimi olar:
t ffi = t ffi = lil =,r' * "
='r1u,*"io') * " =
1ff7 + "
aI4. t e =,.'b'*"'Y
Halli x=orgr orazlemesini dail edek. Onda
r = qct|l almr, Bunlan itoqralda nszcm alsaq:cl
cog., - r.coi_,
4t.'{ry-7'T;d,
\f,"--\t*"ut=\sint+e=
-",g1\rr=\-+t"={}*".-a) o, L2 o" Jo2 *r2,l'*7
an t81,.'\a-xEalli x = q@s2t ovazlamesini aparaq. Buradan & = -Zosin2ldt,
, = 1 r"o* I a[nu. Bunlan itreqralda azoa elsaq:2a-t
= -+atpoez tdt = -2q1fi + ol,ztpt =-\r * rlriozr)*. =
( t I r c--7\=--{tElqrltr- + -14' - x-
)+ c.
116 l,t@-:i[-fu 1t>a1.
11ryi ,-o=(b-a)sin2r avedemesini tetbiq edak. Buradan
*=o*(b - o)ri't, dx=2(b - a)s costdt
alanq. BunJan radikalda nears alsaq:
a=-9;at.Buradran
a--.-dlr oos_l t
'5';14-'I
= -;38
= !i* i. i"-(* "o)].
" = -{"-*" ; -,[- -] -
" =
JGr -;W4 = 1 b - a)sn2 t. [b - a - (b- o)"in2 4 =
= "r/(a - ")ri"2 r . (a - o)*2 r = rle -o)2sin2iLl r = (a - a)sin r cosr.
Alurnlg neticsleri inteqralda nazeo alaq:
J tF -Aff -;f e = 2(b - a12 ! saz t oosz a, = @ -l\' ! tr - * o,v, =
=(o ?' a - !* *1* "
= (' --")',*;"f,- *
*s2j+t9,[G:N4 *"Hiperplik x=qrhr, x=acht va ssira avnlamalar fuil etmaHa
a Sa(tda h inteqra I I an tapm :
u7. I+&."!a' + x'
flalL x=sht evazlemesini aparaq. Bundan &= achrdt, stu=!. t -ni
tapaq. B nr gdm .rfu - nrn melum si1 =et --e-t ifrdesindan istifrda e&k:
e, - e-l x2 =;'
,tu -4", -l=o = ,, =!r,Et > et=x 1!,{j *} -o o lo. o o
=,=u[l*]671' -\a "''
- )Ovozlamani inteqralda nezsra alsaq:
4{ :*-rYnur = o2 ltLnar = o2 1 "t2
at =4a' + a'shtt
= o2 1\1"nzt - rpt = *(i* -,) - "auraaa r=u(r*l .f;A)
EtE IJG;;F;51d,.flolli Burada x + a =(b - a\hzt qr ovszlemacini aparaq,
x---a+(t-a\sn2t diferensiallasaq e=?(b-obhtchtdt alanq.Ovezlomodon
l6t
:1,2r=fi>snt=F"-
- "'-"-' = @ -"o -r-fEr,-1=(, =2 \b-d ID-a
- -,=Ele,EE;r-,--{ E*; L+Dl\b-a lb-o [ia-"
- {a;}x + b = -a + (b - a\h2 t + b =(b - o)+ (b - a\tttz t = (u - aln\.
Aldrpmrz a*iceleri ineqralda nezers alaq:
f[(t -+rt,la- 'b*,'2(b.q\,htchdt =
= z(o - of 1 rn2 t"n2 at =b -iY 1"nx - ty"na + t)at =
= Q:-d- {et, z, - tb t = 9:!- tl4 : - +, =
=(t -d)'1
11enrt - ryt =@ -;Y (l,nu -,). ",
r.*,='(,E.rt*)Hissa-hisso inteqralhrrra suhaun tetbiq etmakle inteqrallan
hesabloyu:t19. Ixn hxtu (nrrl
IIdllL
=4rr- r ,r,a=4ro ,- -)....-r,*L +".nq1" n+l'- n+7 (n+lf
E2A ft2tu2l6.HilIL
L=hr, au=r'4lxot,rtu=l,. |. ,,, l=#t'-*Y'+t
t-dt=
W=:dx. Y=-: II r n+ll
V =, , at =';"z,.dt r ^fr2sin2-!&=l r l= - i r'cos2t +lxcos?t/r=' l2xd'= tlu , v=- rwzxl z
lr=u - dv=mx2xd!=l r l=- lr2*u2t*I";, z"-!rc-zr*=F=du. "=r"-2, | 2 2 2'
= -112 "*2r * l5o 2t+lqszx+c.224
821. ltocqrdx.Ealll l*e4gala|r iki firnksiyadan sctgE nishrt a getin funlcifadr.
Odur ki, hissa-hissa rnteqrallarna ifsdrmu tetbiq ddikdo ocp-i a kimigdt[rmsk lazrmdrr:
F = qrttgx, dv=4lxoc,grh) - 6 ,z l=!-*"tsr-l! "-&=ldt = -------;, v=-- | Z Zl+f
I l+x' z I
= * *"* - lt(' - #Y = * *"* - le - *"w) *, =
=l[,*t\"w-t*,E2Z I sir xlnQgx\b .
Eallip =I"({cr) dv=sinrb I
Isiarhfisrle=|, r & & i=llu=- ' i =- y=-ccrj ,gx c6zr Em.tcoo r I
= -ccx.h(tgr)+ Jce6r, --4- = -cosx .lo(W)* t9 =3m.l@81 'srnr
= -"",,r"trrrt - r'fr jl . "tl23 J(arusinr)'?ar.
Urri (arcsin rF fimksi)ashn ideqra}n tapmaq alar devil. O&, ktonu r kimi gfftrok:
fi = (eru*nrf ,j(arq!io#d=I. I
r=z*u *'1;7e'
- r(arcsin xf - 2Jarcsinx.p=ll- r"
163
d"=e)
,=rl
h*""=''=l#"=*
= <",*rA' -r{- {:7-"*,-,{- ft) =
= ,(,*rir,r1'z - z(- 't[ -/*"i, r * x)+. =
= r("r.r.r)2 * z(frJ *"ior-r)*".
iw"ffir*Ealll R,uradan aydrn gorturff ki, surotdski r-in
imeqralom agilmasrna komek edir. Odur kr hisse-hissa
- -rdtJt -12
un rffiirteqrallama
= - 7*4. It f1 = - ifr4* f,-"* * "
sx tdjrEelll lnqralall kesrde mtoyysn gevirme aparmaq 'laha slveriglidir.
Odur ki,
fu+Ei* )|-* )b"tq-Bu halda vcrilen inteqral rki inteqralrn farqi olur, yeni
t61
(1)
I
o'+r'
+
t, & l. 12 I I r Ij t,\ t _ ;r t
Q;:ie = j. ;*,8 ;_. r
I xdxlx=u tIv=-, =;!_*=l b'.sf
V',,'l 7=* ,=)tb,.tl'rb, *,,)=-)I r l, & I r I x
= -i ;rJ * rt ;i7 = -1 j * s * ;*"tg-
1 -nin bu qiymetini ( l)"da yenne pzsaq, alarq:. & I r t( t x I '\t7;l = }*"re ; -l- 1 pfi * **,ri)
lrt= zssctc;+ 2a2F14+c--aftV826 l=--cb.
It * "'f'Halli Yenlen ifrqrah bir qcder bagqa gahlde yazaq:
lt**Tt=1 '=a[-'wll'-'Y Jr*'2 '
Bundan sonar hisse-hisse inteqrallama Usulurnr tctbiq edsk:I , & I, Er=----, dt =-l
;!:-alr*as)) Jl*'2 h*,'W=r/r+r' V'=ob**)'"=";*' I
={Z-1 '-"\,6=*-Y-t ' dL*,,'l=Jr*? 'fi*,rY, ,t1*'2 ',fi+,z-' I
a"=atg*") n*"@ | r:-;*-1=ffi1*"-*r,ffi*)=ll
l,=l.tl,,=l- rhWu=-----,| 1.,'Y
xldqg eorctg . fr*.,F ,=F-ffi-'1,,41*GOr0ndtyu kimi veribn ideqral yenidcn takrar olundu Odur ki, tanlik
kimi bll etmakle veribn imeqrah alanq:
, o@ - (51) ,-"rg, -I o*"7- , d*,l'-:;d=j'h*tY- 'h*'2 'fr"'Y'
--tcts, G _t) -*"w2l: -e=-7-='in,'Y {r+r'
, rz@s , * = _jl!__e*F +c.'i-.sYi 2"lt+'2
t27. bn(tnxYb.llatli Hisss-hisss fttsqrallama usulusu tabiq edak:
. ll = ern(nr). dv='t!:l_
,srn(hr)ft " ld,
= 1r*r(ror)*,, = * i
=
= rsin (h r) _ rcoe(ra* = [:li rl =,1 =
= rsio(hr)- rco(ln, - J"(t ,)Eh
Bundan sonra xotti tailiyin h,lli kimi2J sin(h r!tu = Isitr(h r) - r"odu '),I sin(b x)rr = ;[Eb(h
x) - cos(u r)] + c
almar.E2& le^ zbbx&fldll
L="*. dv.-su,bx d ,I -les sinbr ib=l ' l=-|eccocDr+,'t. "-^--p=u*tb, v=_i(s,bx I b
l'^ =', ttv = ao'sbx d t+a-1"^ cosbxdr=l t l=-'e*ms6r+b'
V* =*, v= oshbx I
D
* 9(! "* "* t' - ile- sin D' &) =
= -lr* *0,*1"* ,a*-$P* "au
*.Owelki int€qrnl trkrar olundu. xatti lalik hmi hcll etsoh mda alarq:
!e* smbua + *!e* silrbr e =;bsin 6r - DcoDr).
Buradan sada gevirma ile alanq:
!e* w*e=$ ,z(asin6r -DoosDr)+c .
en. !p' -asxf ax.
Ealli $woba di gevirme aparaq:
*1f(t**"zrlt--1"b ' 2-t *!, * !rio2r.2"'224la!1 1 =le'wx& inteqratn tapaq:
ft' -"^'Y*=ftb -2"'*', +*r2 xfu=pba-z1"'*'x *+
t =1"' r*t*=l =o' e=*4=;"-r-Je'sinrd=px&=du, v =smx I
-f=et 'dv=s1'x/, l= "'rior- l"'o*r + je'cosr*)-W=e'&'v=-crsi
=€r gin.r+er cosx- ler qrsfr.Burada xati tenlilm halli kimi Flagsaq, alaflq:
t =!e'mrar.=i$inr+cosr).Bu qiymati (l)da nez:ro alsaq, yazanq:
t[' - -.rf a = ]rk + 1sin2.r - e'(sin, ** r)* 1r r ".
NtL. t:'=&.'(r+lftelli lfre4ralaln ifidenin iizprinde miloy'€r gwirme aparaq. Onda
alanq:re' *_r+t-]",*= "_* _ _!_a'
(r+ lf (r+lf r+l (x+l|
(l)
Aldrpmu axnncr ihdani verilen inteqral da yeim yaa4:
[,o',=&=l t' dr-l "' -&'(r+ lf 'x+1 '(r+lf
,,=l*^, ,,=lG_+*kimi iSara edak. Indi /, itrteqralu hesablamaqla maglul olaq:
. e, l="', a'=(x+tl2/ eX . e,I2=l--: -e=l , r=--:- +t- di.'(,+r)2- P="'*.,=-* | x+t rr+r
1, -nh bu ahnmrg qiymetim (l) beraberliyinde nazera ahq:
tfi*=F-*.*-t**=*Belofikle,
.xz'-e'l-dt=-+c'(r+lf r+lahar.
Eir svo inteqrulhnn hesablanmos la'odtat frghdhntn konontk SaHagatiilnasina ua aSa{tdah dfrsturlara asaslantr:
.&lrI. I --::- =!q7s6! +c (a*o);a'+t' o a
tl. 1--e -=.1 6]'*"1+" 6ro;;A'-x' za E-A)trI j+;=+1lnh2 t r'l+ c;
a'lx' I t
IV. J$=arcsinl+c (d >o);' ,lo2 -r2 o '
(l)
V. 1$=r"1r.,,tr t "'l* " 1o,o;;''Jr'xoz I I
v. l+==*J7*?*" (a>o);',la' t x'
vil. l.[a, -x,*=;,[7 -,, -+**m!+c (o>o);
vfi. flJ tla,=;r-,, * ,lnl".Jj *r1.." (a>o).
s3L t --dx- -'xz -x+2Ealli Kuadral llh€dlinh kanonik Fkile gatirilrnesi qaydasrndan
istifrdo edek
t b =,'(--;)t_.2_,a=t1,_y1rj.\ 2) 4
nuradu , ^ ] =, kimi ovezlomo apiusaq vo yuxandalo I dsturundan2
istiftda asslq almq:,d22t22x-l1,,
"S = fi ."tsli + c=
li*cts -f,T + c.
't[t]$2. I *.
' ra -2/ -lHdtE lj*;=;l# -
Burada x2=r avezlomasiaparsa4:
I . dt t. dh-t\ r h-t-Ji|. I i -' =1[, -v..r = '= hE__:__=!+ "
(l)2't'-2t-t 2'Q-tY -2 aJ2 lt-t+J2lBurada nrxandalo t.e ==
lloi'*"|*" diiLstuundar istifrd.' 'a'-/ 2a lx-oledilmisdir. (l) berabsrliyindo kiihne deyigene qayrdaq. Onda alanq:
. tu r ,lt2-r-.Izlt7 -;7;'7g"lEl;:E1."
$3. I = tu'x" - 2rwq +l
EaIIL ite{f,alala ifadanin itzerinda milayyan gevirmo aporaq:x& (x - cosa) + cos a
7 4r" " -L= G- "*3 -t-*' "* =
x-oosq . cosa
l, -,*;r; "R;d" G;8;il,eBunlan inteqralda mzxa alaq'.
. x-casd I
' G -;;7;'; ctx + cosql --- -; -----.'nx =
t,d[(r-cosaP+sio2ol . --- -, d(x-cosa)= I " i : '+coial
--
-2' (x .-cosaf +sirzo ''--*'(1-coea)z +"n2,
I '- ,. . I dtgl, x -6Sd=:lnl[r - cosaf +siD-al+- utclg -------+ c =2 l' i gnrz - sEa
= 16112 -zr*uo* tl+ cgatagl-9L a ".2l I - srn.!
$4 1--Ig-.'xo -l -2
flalli hnqralaltr i6de iiarind, sade gevirme apad4dan sonra yenidalqon daril e&k:
,o#-, =,,8-l:.<, - ;l =1, - i =
l'"-;J -+ |
=ir
=:'f'-4,'1.* +'[,
I
.!14,=ltt-i
a(,'-(l't , . dt- L-l-=
,z - r2rz ' 6' ,z -r3rz'2- '2'
I
7-t4
=V;=4=rt#=
."=f ul,u -l -d* t J'l-'l *..6 r r re lr'_rl
s3s. r e'sinx+2ooor+3
Ealll s;ar vo codr fitnksiyalafll yanmarqrunoltla ifada edenlq kssribiroins funtsiya qoklkto salaq:
zanlcel+ zax2 l- z*n2 |+z*2 ! * rr' i
.&= t--'
z"io Icos I + 5cos2I + rio2I2222Brmdan soua kasrin surot va msxrccini o*' j-re bolsk:
1&/,.\-s2r al Ell
t 2 =zl \z)t
w? *,i ;='t,t' lt rrrli 5
l?0
I
=i'-*,=4
dx
= urA;-!- =, lo*ol = *"/i! * "
aa n*-a*.{x'+r+1
,IIaIii, M$eyyen sadalegdirma apar:aq:
al,, * 1)*1rl:4={r*;*l2'k r\2 3 2
i['*l] *;$2. l=&-*
Jl+sinr+cos"rHelll
al1 -,'l I
= -lE+ =-,*"u{ * " =,,*r.'' ;t *
" =
.12 _[1-,]- ,J/+ (z '/
. 2siar - I=arcsm--+c.
s3s. t--!:!----n,.'Jl+x2 -x'ilrrA fuile.qralah ifrdo uzerinde sadc aevirmc apanq:
t*13#fr=Jt**2-14
a=ls,Br=4=l--=',12 +, - t'
- 1l l-----*. t# = !.tbz *, *rllalz *,*r),. ix"+x+l V-ts'+E+l "
. d(siur) , d(!in r),--.------rt:-'Jl +sinx+ l-sin2r 'J2+silr-sia2x
&+
t7l
. .r+1 L ?: b l, ?-x+l| _::.= t =4-Lt : =- L_ra'l?+x+t 2"1]+r*r
"l*+x+r z"li*r*t
- L -4!- = -1 --!!- - t( I -,r\-L' r( t -,r1413 t 413 z 4\4 / \4 )lo-' li-"Bu axrncr ifadeleri irteqralda nazors alaq. Onda
I
-3r L -!{1 -,2)- z{ 1 -,2\=-1,,*i"4 - I4'l3 r 4'\4 , \4 ) 4 J3 2
i; -'^,I!-")3 l\2 I I=-4*.--T--'2 ,-
=-l"r""irr-zj -!hrt2 -14 *".a Jt 2'
h^l;-t'+c=
,//s. H+:x. rlx, +x_l
EaUi Bu up imeqallann taplrnasr
istifrde eunak lazrmdr. Difetensiallq saq
iteqalda nszsro alaq:I
. tz tdt 1,1-A-l l. l-At____+: _-_-__:4.= - t __:,1, _'t /t *!-r 'Jr*r-r2 2',,h*t- t2 2',fia1-tz'P\P I
'(j -,tr7-Ttl;-lr-'J
| .l-zt I x-2+ -arosn-+C= ll +- - .- +-afasln ._+c2 .J5 I , ,2 2 ,15.*
iig'* r=1 evszlemosindsn
*=-)" ahnq. Buntan
=^lt*,u'*
I+-2
840. t e _.'(r-tlJr']-z
Burada r-r=l evezlemasini aparaq. OndaI
ahnar. Bun-lan irteqralda nszere alaq:
!a,
jfrq=dt
=-t-=l,h+2t-r'
l-l r-2= altsllI.- + C = arCEll __:;--__- -lC.
J2 J2(r - I)
,lz*t*t'a.Kvadrat udoedhni kanonik gakle gstirsk.
Bu inleqralm tapilmast ffgun
t= I vor-lHallL
a, = -)at1'
-t I
ld,t2 a(t-t) _ffi-1
I
Ml.lHolll
+ E+. - t *It E.,..,[;;71. " =
J+ l r;;?. ]r"l] -,..6 -,.71. "
173
yuTJ * = i.l7;7 +++..L';e. "ccdrcl inteqralmdan i$ifide etsok, alarq:
I
Halli lntqralalt;r ifadsnin suret ve moxrecini r2-na bdlak.Il+ ^
=t
Mz t4:u.rr/r' + I
Burada r-!=r ila isara edak. Ondar
l" lli''-Jl
w.,Jg+ o.p,..xr/l+r-x'
flrrri Intoqral altDdah kosn iki kosnn csmi geklinda yazaq:
r-{-L:, '(r-!I.-L=r''(,vi;7 .'f.,-75*-r'ffi-+!-;:L=48=\+t2
nil+r-x"Ow€lce {-i tapaq:
J, =J
, = ! orazlemoshdc , t = -)at ahnq. Bunhn iueqralda nezare alaq:
-I*',=j, T=?T
;l'*i- 7
.dt,Jl +t -l
,(, *!), i 2l=-t---\.:-
rf I\' s
{('.;.J -;= _"1, . I _ r,;;l = *l: . i .,8+-;l,
- it # = -)fi .' - sli r[ t. -.,)* lt
--,'[*-J-l".,i,,f +".
ltva I2-nia qiymetlerini (l) <le yerins p256q a5pq;
r = 11 + r 2 = sfi +* - ffi -.,[;; - ;",o. I + *,
(D
44pa,{l+r-r'
f. ztb &2'$+'-t ',ll;-,'-. d l- l-2x-tgg=-1-d-l-2x-!.
o(\-,\(2 lls rr 12
{; - l; -'J
l.=- rl
171
Rasionalfunlaiylann inteqralr tapm:
w* t1;1 qI/aIIi Namalum emsallar daxil cdak:
xABC(r+tlr+2)(r+3) r+ I r+2 r+3
- z[]* s, + o)* a(r2 + lr+ r)+ c(t2 * l' * z)
(.r+l[r+2[x+3)
r=_L;4.u4_rr,,_ =
=-]up*rl* zrop+ zl - llx+ 3l+c="Efffi; -"a
64s. r=1,--#Halli ifrr4pal altmdah kosr dEzgun ohayan kesrdir. Onu cwelce
dtzgun kesr poklina g*irok:,o =, 5xz +4
7;;7;-',-7;;7./'Bu halda r futeqra[ iki iqt€qrah farqiden iberst olar:
'=r['-;,;*:)*=w'Q!] .*=
- 5r2+{=r-l ,---- ^-G=x-tl'r'+5.r'+4
Indi .lr inteqrdto hesablayaq:
- 5r2 +4I=l-dt.| 'r'+5x'+4Kesrin msrocini vuruqlara ayraq 14 +5x2 +4=0, 12 =r ile igao etsak,
alanq:
lA+B+c=0, lc=-(e+B\ lc=-le+a\ V=-;'.{u * or.:c =,, = ls,t*as-z,t-to=\>]zt+a=\ =la=2,le.t +ta,zc =o lat+n-ztt-u=o l4A+B=o
t.=_;
115
tz +5t+4=0,
-5x Jz5-t6 -sr3t=-z-= , , ,r=-l,lz=-4.
x4 +sxz +4=Q +lru +4=b2 *rl'2 *a).Bunlan 1,- da nezsre alaq:
- 5x2 +4tt=tg;6444x.Imeqratatu kesri sade kosrleo ayuaq:
5x2 +4 Ax+B Cx+ D
A=0,
B=_1.?'
C =0,
o=16.3
l. c& 16. & I t xtr =- i7]l+Tl S * n=-1*av '-sctg- + c.
l, -in bu qiymotini 1-nin ifadesinde yerina yazsa4, alanq:
I=v-1 = v-Lq's11 t rI I fE + -dtctSr+ c '
t46 t=tl ,' )e.\x' -3x + 2)
EaUi lftr4ralallr kesrin maxracini vuruqlara ayraq:
xz -tx+z=(r_t\x_z).
[a+c =0,ln + o--s.
laA+C =o,
l+a+o=t
Bu hlda tnteoral t =;-!----=a kimi olar. lndi -j=__'(,-rf(,-212-' -'- (r-tF(x-2F
kesrini sado kosrlen ayffirq:
176
llt * +'l+ ok2 + +)+chr + xf + a[,2 + t2 + lb2 + +
X2ABCD(.r-t)'z("-zf -r-l (x-lf '*-2'G-i.;/-
_ e(x - r\x - zf + a(x - zY + c (r - rP G - zl + 4x - rY(x-f (,-zf
,2 =,q(l -s,2 *a,-t)*t('2 -+,*t)*c(i -u2 *s,-r)*oQ, -r**r)
, = q*. t& - tf. tf,=enl, -rl- J. - rr,l,- 4-
- +-! * "=6fl. 45. ;..= o"fil _ _r11..*'tabaEalli htcqral alhndak kasri namelum emsallaru komeyi ile sada
kosrlore aytaq:I A b+C
Glffi=;*,n,=
[A+c =0, F=-o. lc =_A. lA=4.)-54+B-4C+D=t, . l-A+B+D=t, lO=ze-ct la=r.lae -
te, r - zo = o,- ft,t - +e - zo = o,= le - * = t,' ='ic =*,l- 4A+ 48 - 2C + D=0 l-zt+ta+D=o l_e*ta=o Lr=o
ft+a=0. lr=-o,4B +C =0, > |C=-B=L -l,r.*c=r.
l^=:ll
r=)t4.t\]*=j+.,t=irp.,t - inl,,. \*l-,w *
".
eee r lft,
l|z +t
tn
c(,
*1r2',
I/allr Inteqralaltr kasrin mexraoini rtruqlara ayrmaqla msional kesn
sade gsHa salaq:
1 I A Bx+C
,5- U-.rF- x+t)-:+l '2 -'+t
l,q.+ B =o. lB=-A,)- o*, t, =0,-l-zt*c =0.-[.a*c =r [-'r*c =r
A=\.3
I
I,3
t2,-1 , & *,-3r+1&=Lulr+rl- !p2]- *-!-4-=' -
JJ r * t ' r *2 _ r * l- 3' ' 6'ru -5+l 5-a'-x+l
= J r,r.' - | rrJr_ -1-+ - [t f;.'tt l*=r 1 .aL'-*,,1 '('-1)
= ;rnpr+
rr- e,--- -l- - i, j#, =
['-rJ *;
= f r,t, - 11 - |ul,' - * * t\* $**s2!! *
" =
--"P:.4a,asff+".1/,2 -r+ I {l
ue. r lfiIlalli Owele inteqralaltr kesrin moxrocini wruqlara alrraq:
"4-,=(,,.',f - z'2 =Qz -J1**,1,'*Jr,*,),
Bunu inteqralda nezaro alsaq:
F-n:,Y.t;alndi imoqratr saaa kesrlaro ayrma iisulunun kom'yi ila hell edsk:
1 Ax+B Cx+ D
V:ilV;i;l= ,t-- rz*i ,2. Jr'.I
.tlx2 -, +t)+ ar2 + a' { C, + c
t= a(i ,.fz*2*,)* nQ' t 'tz,*l*cl*3 -.1x2*)t oL'2 -,fz**rl
1
2.t2
-l,
^l- 2i1'
o=r-l=1Taprlan smsallann k6oef i ils inteqral agagdakr gokle diiqar:
llll, = tEll* - t Fi_L * = - #t7ft^* -
1. dt I r . l. &* ilEJr, *r* lFrl ,1sr,*r*'ij r, *,?r*, =
t 2x-Ji l. dx l, &fi - -1 _;--- +; j --;----;:- r
4.12'x2 -Jir+t 4'xz -J2*+t 2'xz -Jlr+tt - 2x+Jl l. & l. &-;fr! , ^, -r,-,*-ir,\.r2,*r+il x\{zx+r=
lA+c =0, [c =-A,l.[zt*s_.fzc, o=o- lo=r -a.J -{- - =ll+ Jla rc - r5.o=0. l.Jze+ I + Jze+t - s =o.II[a*o=r l,t*^lza-,n-Jlr,fza=o
t-'lD=t- B,
-l.-=i21u=-1.lz^tia = Ji
l"I .dk" - l2r +'l= _- I _---I-
4J1' ,2 - -2, +l.+,-tfl.;1##.
lx-- I +-trt,t\ --l
t' t;\- tllx+-lli-t, -l- z ) r .t z_g,,1*z:!2_",,{r,-t*. ,AJ- ofl--iJrtall .-- '.t, l, *-.b r '
lI+-l +-I 2./ 2
t19
- # *",, +, firop' *,rr, * 11= ;^ftH,-'if*"o7,-l*fff-"e (J-2, + r)+ c =
=#'l#|#-"oi."
16+t
,1 -12 +l+ 12=fr4{:7A+t*=rF.r).rF'A-f:;;;;1
Brmlrn verilen imeqrdaa nezzilc a14- Ooda verilen i*eqral 3 itru1rdtncsmirden ibarst olar:
, = 1r fi. it #,- it fj;i = r 1 + r 2 - L 4
IIer tir imeqrah aynltqda taPaq:
- r- & Itt=rlZi=roctcx,
,, = y ft=! ffi,=v = 4= I' h=! *,8, = !-ad.
,,4jln4fu
(r-r'[*,'l2 +tlra - 12 +t
x2 -l
*2 -1 Ax+ B Cx+D
+.f:r + t
It{)
A=-+.
s=-I.2'
.= |- Jt'D=-l
2ll ll
h=r -J=e,t,$'
=i *= -];t-4-' -" 'xz+Jix+l 'Yz -^f3x+l J3'12+J'r+lt, & l. xd t. & t .zx+Ji-^ll*_- rr rf + Jr'; Et7: Jk + t- rt 7 - J\x n= -*t;z*E+
I1, dr I .2x-11+Jl . t. &- it ,2 *{3,*t* 2ht; -yf,;f - rtj -;73,-t=
-- r ,a(': *.0,*rl*li_^-4--l r= { -.+J-rdb-':
-Jt +r)*2J3' ,2 *"13r11 2'? +.flx+t 2'12 *.f1r+l' iJtt ? J1rrl -
. it 7#; - :t
"+; = - rTrL"l'2
* .rr, * rl' }'1" - r:, * rl =
_ _l 61,2 - tr:, * rl2^13 lxz +.fix + tl
Alrnmq i6do1r6 7 =^+12-:h -de neare alsaq, iuteqraft tapmf olanq:
, = t * = i_" *. i*" d _ h41fd,,].,1. "
ast t =t -- .! , =IIellL Owabe inteqralalt kesrin mexrocini wruqlara ayraq va sade
kesrlor gaklinde yazaq:
fa+c =0, lc =-e,ttl- Jtet a * Jx: + o=L lD=-l - B,
1 t-,rEa *c, Jio=0. =l- S; * a -,fi,q -r - a =r,)[r*o=-r le-l1s-t-lr-^fia=o
(C =_A,
lo=-t-a,l-2-l3A= 2,
l-z.t-ta = ll
15 -14 +i -x? +r-l=xa(r-l)+r'z(r-t)+(x-t)=
=1,-r{ro * r' * t)=1, -t{r' *'+ t[,'? - -*t)
I A Br+C Lx+ E
G_sr;;f; n='- i * ;1,+l *
7 -;'
I
3
8=_1.3'
_l
r=.t(a +J +r)+aQo -2" *2,'-')* .
*c(i - 2,2 * z, -r)* z,(" - J. r(" - r)
fe+t+o=0,l- za, c * r =0,I,A+?ts -2C =0,
l-a.zc-o=0.le-c - n =t
+4C =-:6'D =0,
I
2
lllr=11e *lir-Z**t 1 ,x--
3'r-l 'rl +r+1 'xt +x+l
=ir"e-rr-i1{it,, t * -!fi l7*,== j'r, -,r- ii@ .!tf;- yt* i,S+ =
\ 2i 4
= irl. - I - *tl,'.,. !- $*",szff . "
=
ffil +-"82#*"
lE2
x2&
xa +3*1 +2i +3x+t2
Halll Owalco kasrin mexracini vuruqlara ayrraq. Bunaim€qralalh ihd'nin surot ve mex.cini 2-ya wraq. Onda inteqral
,2d. , 2?e' -J
o -,.Lr*3r*,
- t;\;;ig.\;x+22
852 I =l
qeklina dugar.
oldugundaa2x' +6x! +9i +6x+z=lJ +zr +z\zl +zx+t)
Ar+B Cx+D2+2xt2br2+2x+ll 12 +2x+2 2x2+2x+l
2,2 =eb,3,z'2 +)-lr(rS +z*+t)+c!3 +2,2 *z*)*o(,2 *2,+z)
lzA+c =o.
l2A+28+2C+D=2,<-lA
+ 28 + 2C +2D =0.
lB+2D=0
C = -2A-I
la=-zo.(-l2A-aD-4A+D=2,l,t - to - tl +zo =o
f
o=^9.5
_85'
B=12.5
Bunlan verilen iflteqralda nazem alsaq:417t6-r+- --x--t=!#*4+e=:;J9-*
x'+2x+2 2r'+2r+l > x'+2x+212. & 2. 4x&
r;J-;--:l--;---) x" +2x+2 I 2x'+2x+l6, d\ 2. 2x+2-2 12. &
f '-ttur=) _---
5' zt2 +2r*l 5' I +2r+2 5' xt +2x+22,4x+2-2 6. & 2. 2t+2 8- &- it 1,4, - ax - IJF;r;r =it?
* 2, * ;* *il?. 2,, 2-
lc = -24I B = --2D.>{=l-2A- 1D - 2.
l,-tt-zo=o
163
212
2- 4x+2 2. &--t._;..',----.- '- -tz---t__;-=) 7x'.2x+l ) 2x'+2x+l
-,a(,2 *zr*zl .t, a(r+r'l -z,abx2 +zx+r\-,, '('.i) -
'x2*2r+2 5'(5+tf *t 5' 2x2+2r+t t'(r*1.), *l\ 2) 4
=lt"l,2 *2. * zl*l*986*t)-llz"2 ,2, +tl-?*as2'-+1 * " =
= 2- yrl
".* z" zl * l *"EG + i - ?- *"qg(2r + l) + cs l2x'+2x+ll ) )
Ostroqradslri dasnrunu tatbiq etme a inteqrallon lapm:
ts3. r=t * =.'(r-tf(x+lfI{aIIi Osroqradski nezariyyasinc osasan
E=-D,
A=2D,
2D-28+2D=0, +
-4D+B-3C-2D=1,C-B-D_D=O
lB=2D, lc = 4D,
- |-rn * zo - { - 2D = t,) I B = 2D,
lc -ro-ro=o l-ia-rzo=,o=-f. r=1. ,t=-!. a=-!. c=-L16 16 8 8 4
It4
t dr _Axz+Bx+C *o, e *tte ,)'(x-rf(r+rf (x-llx+lf 't-l 'x+1
yazanq. Indi boobarliyin her tersfiniri diferensiallasaq, alanqr. -,,-V.-- ,r ^
', -(z*idv'.i-l*'*a,*cb*-i+nr+C[3x-l) . D E
1r-y1'*-- G-r)'?G+rf .r-l'r+l'Sa[ tcrafi ortaq mexraca getirssk:
x= -*3 +(A-zBhz +(-zA+ B'sc\x +c - a +
* o(r- rl/ +:r2 +r,+t)+r(rn -2"2 *r)D+E=0,
- A+2D=0,
A-28-2E=O, =- 2A+ B -3C -2D =1,
C-B-D+E=O
Omsallann hr qiymstlarini (1) betabertyrnde nezars alaq:lr I I
r d =-e^ -s'-a _t_tL*!t!'=
'(r-tP(r*rP (r-l)(x+l)2 l6'r-l l6'r+1
-- x2+x*2 -llolr- +lhlr+ll+c=8(r-llr+t)2 16 ' t6 '
,2+r+2 l.lr+ll=__t _lnF__l+ c!
s(x-r(r+t)z 16 lx-ll
ssL. I- I -t'*t[IIallL ltqrahlttkesrin menocini vuruqlara apraq:
.&&tGt;il1G.,rF;TBu bteqrdla Ostroqra6ki usulunu tatbiq edek:
. dt Ax2+Bt+C ^, & , ful- Ftr*ff;;g=8ff:.o1-'t,* (r)
Axnncr boraberliyi diferensiallasaq, alanq:
l=-Ax4 -, zh,,3 -cr2 +z*+e+olxs -ra *rt ,12 - r*l)** EQs *t3 *12 * r)*r("a *12 *, * r).
C =0,
a =!.3'A=o,
o=?.9'
r=-?.9'
D+E=0,-A-D+E+F=O,-28 + D+F =0, ?-tC+D+E=0,2A-D+E+F=0,B+D+F=l
-A-2D+F=0,-28+D+F=0, -c =0,2A-2D+F=O,B+D+F=1
p=!9
It5
Nanplurn omsallann bu qiymatlerini (I) boraberlMnde nazera alsaq,alanq:
11.4, cb it 2, dt --r+-iffi=;;.;r#1,t+--:*=
= ffi . Jnr
-, v it}*,*.it j*,==t';1.f't'.,t-!t$iS ,l." - t')l. \ 2)+:J-=) ( l\' 3l.r--l +-\ 2) 4
= t', ;. Jtt'.,1 - ]r"1,' -, * ri * 4 -",sff * "
=
x l. (x+l)2 2 2r-l= fi, ;l). r'F; + ----F qtts
-6- + c'
tss.1 ., & ,, .
Fo*tf.EaIIi O*roqradski iisuluna esasan yazanq:
. & Ai+Bx2+Cx+D ,EJ +Fx2 +Mx+N -
'!F;f=--lt.r +t--7-- --e
llsr orrfi diferensiallasaq, alanq:
t $*2 + zax +cLa * r)-+rr(zrl + Bx2 +cxrDlF;f=E'
(l)
Ex3 + Fx2 +Mr + N' t4+lt=3Ax6 + 2Bxs +Cx1 +3,4x2 +zBx+C-qAr6 - 4Brs - 4Cx4 - 4Dx1 +
*(ra ,tlgrYr * rr2 +ur+ w),
l=-Ax6 -28x5 -{*4 - ADrl +3At2 +2Dx+c +
+ b7 + Fx6 +Mr5 + Nx4 + Ei + Fx2 + Mr + N.
E =0,-A+F=0,-28 +M =0,_f+lI=0, +-4D+E=O,3A + l- =0,28+M=0,C+N=1
B=0, D=o, E =0,
u=0. c=!. N=!ln=o'[r =0'
Bu qif,metlori (l){e Dezere alaq:. d, x 1- &
'1.r.7=6.4"+rF,,'
s* *;;*l.u.,'flalli Owelcc imqralaltr kssrin maxrscini wruqlara ayraq:
x4 +2i + _7x2 +2xrt=(r2 * r*tl .
Onda irteqfat asa$dah kimi olar:
.&&t i;;\ 3x\ zr + t=
I
6, .,;7 t
Ahnrn$ inteqrala hoqadski [sulunu tatbiq edak:
. dx Ar+B . Cx+DJ,----------=. = --;- r l -;--.I
P +r+tf " +*+t x" +x+l
( t ) boraberliyini drfercnsiallayaq:
- zh2 + ,. r)- (z' + rXa + a) , Cx+p
r;;,r =*-T;r--7.-*.,'
. dr r 3 ,ltz+$*+tl I J-zxt p; =
6.;1. raEr"lT- rr.;i + " "= qctt ---- + c'
l=Axz -2.8x + A- B +Cx1 +Cx2 +Cx+ Dx2 +Dx+D,
(14)
Ir7
llnir,o=ol-l-28
+C + D=o,
lA- B + D=t
D=A,C =0,
tr=23;,3=l a =2
A-Ltr,s.=t2
a=2.3'
D=!.3
C =0,
^l3
Omsalhnn bu qiymetlerini (t){a nezere alsaq, alaflq:21 2-r+-, dx 1 j .t _3 dx_
'F;=Z-.^ ''! x2 +x+t
/ r\2r+t 2. 4,'*r) zx+t 4 2x+t=p;;1. ;t;ffi = v;1. ra@*g rr + c
\ 2) 4
Mtlxtalif asullart tatbiq etnokla aiabdah inteqrallan tapm.
ss?.1;zIIallL Sale grvirmo aparaq:
'fi1'ffi=iiH##ar=itP-l.dtx2\ t,t r l- dt l- dt-;J,.
_l =1r'=11=l,z _r- il ,\t=
r lr-ll I t- lr2 -rl= i^lfll- i-"*- "
=,1"1fi1 - o1 -",*' .
"
es* t t'& .
'*E+3.Ualli Sade gevirmsden sonn yazaflq:
' ff =;'#,=F' = 4=i' *= \,",s f; *
" =
lx4=_orclg-+C.4,J3 ' ,17
tt8
$9, l'2,+ r&.'.16+l
EaUL lnqrah iki inteqraln cemi kimi ,azaq, bu-dan awelkimisallann bslli gydasmdan istifrda edok'
tfa-q p=:'H,-;'H==i*.*',yH=I*"d.i,
lnai I iteqntrnr agaq:
,tfi=i,'=,|=tfi=t*r,paI A Bt +c Ab2't+tl+Brz*Bt*cr+C
Gil-, ;r=,. r -
;, -; ;i =
- (,. r*t; il--lt+ B=o. lr=-o,l- t* s *c =o,-),c =2,t"
[,r *c =r [r=j.r=_i,.=i
t2, = it fl . r +4= i'r.-,r - irf* a - lt"!^.'zrt f;==j'r.,r-ii$;i)-i,,*l=
't'- r.,) *
a
= ltf .l-*"F' -, *rl* !r'*,zff =
= irl,' .,|-*ti' o -
" * rl* lr-",s2ff
Bu qiymati /-nin ifadcsinde yerine ya.aa4, drq:
tl\ a =!*d-:tll.,1-;+ a',2 *rl* ft*"o#. "'
It9
4-t6t.l#1t;le.Urfli tdeqrdattr fiDksiya [eerirda sada bir gevinne aparsaq, alanq:
, = it 7*G,)=lr = l=it f., .-,t,
Aluruq inteqnl sada kesrlrrre ayrmanrn kdmeyi ilo hell edsk:t-3t-3ABC
Beleliklo, inteqral agagdakr kimi hemblanr:
, = i' FiA" = il-'r",. n * - )' i,j== |f-]'+r. ott'. rl -ir"l'. zl *'] =
= |l- jr"l,'l. +r"1,' .'l - i"l,o .' ril.
"
t6l .t."'t' xa + 3x4 +2
Halll itsqraln tverindo sada gevirme aparsaq, alanq :
i'ffi=1t =,i=|r!t- - i(, 4!-1, =
1 l. 3t+2 1 3. t L dt= q'-il ,l*3uz''=i'- cl ,Ti, *zo'- zl ,\1,*l=
I 3,2t+3-3 I . dt I=-t- l-dr --l- =-t-4 8'i +3t+2 2'/ +3t+2 4
1=-1.2'(,t+t+c=o,
ll.l r lB *c =t,=lze= -t
{. 3
l =-i,)a=a
[._;
,u *" =!).-
t+c=12
190
t . aL' *v *zl s . dt--E)
F i;* E F*3t*2 !-4 --' -|r,i,'*r,*r1'
-'1't|=1-Jr'l".a.:1.I i'l#I."=[,., -; '', l'*'z*11
= | - f r,l,' * r, . zl - itl#l. "
='*1 - lr"l,s * r,a . 4. ;
r]#l. "
-2h-l862.1r,;fIlalli $b elemeotar gevimrs apannaqla verilon imeqrah aga$rdakt
kimi yaaq:
t tfl =V, = 4= ),tf = )\, -j,),, =,tt -,+ *l)* " =
= j("-tl"-'l)."'Ms.1):!-a, .u,rro
r1l+r')I/alli Sada gevirma aparsaq alarq:
)' * = i'() - *Y - i' * = i'i - i' * -
-\ d' =\aut-2-lnll + ri+ c =1'l+t 7 " 'l
= lr,'lrTi -lult* r'i*"= ln- ",=l* ".i*t- | 7-'i' 't ,l(,*,rll
-2 -tE6a.;-1:-a.r'+r'+l
I/a!/r Kosrin surat va maxrscini x2-na bcilek;
t9l
_ll+-i
'rry,f,- l)
. l. x)or
,&=|#1*o^l#1*I - lra - J2xt +ll
= +Tzlr;afi."
iliat=f -;=tl=t7J=
I
= i**i- " = i**r?, " = i*n ff *
"
*'.tfu.Ealli Ksrin suretini sadolosdirek:
,fl* = 1r{* V)=V = 4= ;ria, =, *, =
ffi,r*n.*l*.'16+l
Ealll l$sr1a mcxrccini luruqlara aytaq ve miieyydr elementargevirme aparaq:
. ,4 -x2*l-x2 . & . 12 -I F,;il4= -;t.f = ! -r- + t - E- dc =
'' #-'*'**l*"'gs *"'= arctgx + 1rTCI*t 5
irrasi onal futt*r iylarrn inte qmllorxttast :
s7. 1 x3J?- !l-+.
'x+{2+xtCtL Bwanl- 2+ x = tl avezT*nesini 4arsaq, irrasionallqdan azd
olanq:
r=P -2, &=3lar, r=1Jl+r,.Bunlan inteqraltla rozar: alaq.
1l!,,'a, =t1 ffi = rr(,' -,. ffi\t, =t-to -),2 *
3tz -6t A Bt +C
G1trr.,;)=;* t\t*2'* -a = tl2 + r + z)+ 1ar +c;(r - t!
3t2 -6t 1/ -1? *r.(t -l[t2 +r+2) 42
1A=-:.4
B=15-4'1t- - _=',
lA+B=t,1A-B+c =-6, -l2A-c =o
1532- it fi. tffia, = -1ar - \*l t17" ='i'+ -,t.
. i t ffia - 3rt
7j;a = - iw - 4 - t1 t4jY? a -
T';fur=-i't,-r.fIt+
2 ,t *zl-!!.L*"p-]=I t ./7 " .17
2T=-3-r"t -l*Il/ *r*21- 77--*",rb !1 *".' El | 4J7 " J7
t A+ * Le = -l,,.lr-rl* !!12 +r + zl- 27= *"e!l * ".'r+1Jli 4 , ' tt r +J7 " .ll
Blrlzdln t =Wi.s6B.t--:9==:-=---t-'3{r+rf
1'-r1afl oilL lnrqrala}r ifadeds sadotegdirme ryaraq:
.&J-=---,,,_,r,n[(;-'rI
nr*au {=l evedemesini aparaq.
x+1= rt3 -t3,
,['-,)=,' *, = ,=,Ur_,
o -2,'(3 - i- l:lt' * rl a, = - d'd,,=.
F -'I F -'I
u_,,,=[r.l_,), =frAlman axrflnq troticaleri (l) imeqralmda nazers alaq:
6t7
t $17, = -\ta, = -!r,. " = -1ffi . "
-
r;T'
(t)
E6e.t
olliapaftq:
(n-utural adsddir).
Brmdan swelkr irteqralda oldu[u kimi mueyyol sadalesdirme
dx
x-b =P evazhmesini aporaq:x-o
(1)
x - o -(x -b) * _ rto-tdr t u - o _, *= rrn-r^G-'Y G-oY
Bunlan irt€qralda nszore alaq:
\11*-"y*t1r-of4
r-r)'-l;)
194
s?0. !+=-e{l+r+x'
Halll Mwryan sadale$irme aparaq:
t J9=* = l:::l:Y:* = 5[r,., t * -r,/l+r+r" {l+r+r'
tjl+ r+ 12
- lr,t * I *.trr *ll7 - ^l;;;72l2 I
Axnncr boraborlikde birinci ideqphtr hesablannusrnda agaSrdah hazrdiisturdar istifrda e6k'
2,flJ t "' a. = i"F t,' rli,. l7;J1
Bunm iigiln '=*. '=,.lgei:rsatq alanq:
- ir"|,.i..F.,. r -,tt;i =i# ^ll .,. r -
-lr"l,. j..[,.,.r1.".l9J
-I
,(,.N r
I
l l't +, + r21- i 4Q *, t,z) *
l* =2,:lrf, *,*r*ibt*1*,.,f, *r;-J*r*r2 4 8 I 2 I
871. I
Halli l - x =! avazlemasinit
verilen irnoqralda nozers alsaq:
apamq. Onda e.=\at olar. Bwlart
lo,1',ry- tdt
- t-Ji=l
= :('. 4). " = ;[.a-i. i Jd:F). "
=
.;'(=T
aparaq:
J."={E.ifiS]."
+2!-$-=11+u2,xlx" +2x+2
x
l_2r
d(r + t)
!i]:4*zr@:)tt:4-,,1?+2r,2 'Jft+rf +r 2' J'2 +2,+2
$$ * zd,, w,[7 *:,*zl=,t,' . x *{(r+rf +r I
-r"1,*r.#.2{** zr"l,*r* Jl * z,Jl *
" =,[? *iJ, Ay **,[t *i*j*
"
I
E72 . l::-:::: =dxT
flalli Sana gentme
t=r!:2:i]--*' r,li +?x+2
r"=t-J:=t-=! *=-\*l=t' ',,1r2+2x+2 | t l
l.lr r f]-rl=-E'E*r*il,, *;*rl
1 = 1 r, 21, =,[illli. r"l..- I -.,f,t- z'. zj -
,17;r,.rl" l*", =J*'*or*" lsadrat nghodli olduqda
f--o@6=9,-1Sr*114dtshrundaa istifida oderak irteqrallan tapm. Burada c,(x)-r,g,-1ft) ise (n - l) daroceli eoxhcdli 2 iso edaddtu.
ffi.r&',ll+2r-x1
IIaIlt Yuxanda g6stsrilen dtishrdan istifida edak:1,l+-+=lwz *a'*clll,z*-l? + )t L
,ll+zt - x' ',ll*2*-r2
Boraberliyi diferensiallayaq ve orhq mcxrsce getirak.
(t)
-3
-
+\-xldr2 +Bx+c).4ffi=a^+Bflt+2'-x2 rt*2r-rz .rr*zr-,2i
" (z* + afi + z' - i)* g -,1*2, x * c) * t,
x7 *2Ax+ 4Axz -2Ai + B+2Bx- Bx2 + # + Bx+C -Ai - Bx2 -Cr+ t,
-6
IJ2
I
3
5B=--,
- l96'
Sabitlerin bu qiymetlarini
l=4.1){e nezere alsaq, tazilrq:
'T*?1 :.'-2.-2),ri:i .+ tr-:L-:=tzx2 *sx+w)Gir] *q 44--=',h-2, -t2 {2 -(x -tf= 1212 * 5r*rl1l[ll, --r2 + larcsinf *"'
tzl .1*a.,F -?a* urptr..
Ilalti Sado gevirme aparaq:
1,',F:?*=fS!)a=l++-----= r-1 ;
l-u=t.lrn -r, =0.
i2A +38 -c =0,
lB +C + A=0
(1)
=(*t.*a*o rci ,6*z *6*P'l{o2 } -r.l#_iBaraborlivr diferensialladrqdan sorra" ortaq meEece- gstirsak, alanq:
ozra - ,o =10*i + 4Bxr + 3Cr2 * zo* + tlg2 - ,2)-
- ,(^t * BJo *c"' + Lx2 + Ex + F\+ L,
^a'- 24'
E=-d16'5
"a16
Amsailann tapftflrq qilmetlerini (l) berabsrliyindc nszsra alaq:
-6A= -1,
- 58 =0,
5a2A- 4C = o2,
4Ba2 -3D=0, =s
3Cs2 -28 =0,
2Da2 - F =o,
Eaz + ),=0
a=!.6
B =0,
D=0,
F=0,
l9t
.,'-, ,. (r5 o2 r1 o4r\ I ;-" o6. &lx'{o'-r'ilt=l * l(a---r_+-l:=[ 5 24 t6 )' t6' ,l a2 _x2
( 15 o2r3 o4r'\ t , - ., a6 x=l - --..-.- l.va- -f- + -arcsm-+c.lt6 24 t6) t6 a
87s. r =t--9-.' ,3'lr2 ttHolll Btnla ,=!- *=-\* evezlemasini aparaq ve bu qiymetleri
lt'inteqralda nszero alaq :
I
- dx --2dt - P' i,l,2,t , r 11 _, ,
.,h*,2rr Vr2
'^
Burada yuxandah iisulu atbiq efinek olar:1
!-!-nt =(,* + ail.* t2 + t4 -L, -vl+r' Vl+r'Bu boraberliyi diferensi "
Ilaca rL alanq:
'!- = ^"Gt
*' l4!+B). t -!=',11+t' r/l+t' !I+r-Berabarliln sa! tarefini ortaq mo)(roco gatirck:
t2=A+At2+At2 +Et+1,,
Bu qiymstlsri t#, inteqralun (1) aynlrgrnda nszara alaq:
. 12 - r r I l- dt L r--- l-tl-r::=dt = ; rll+r' -., : = -Ivl + t- --lnl, +'JI*rz 2 t. lt+t2 2' 2l
K6hrn dayigsns qaytsaq, alanq:
(l)
{2A=1, t^=:'lr=n ={a=0,l,t+t=o lr=_IL2
l+t2
v=,[Ji; oltuqda P# rasiotul 4anl,,}'asmt sd'
kasrlara ryrmoqla IW inreqrallannt tapm:
nar =t 4--.' (x -r)2 .'lt + zr - 12
aAir -G-t)+t I I
Gt r= ft-r=,-l.(,;Paynhgmdaa istiftda ersok verilaa imerqSl iki inteqraln ccmindan ibarat
olar:
, -, tr =t 4__+t_____4:=' -' G - rf l[* zr J' (r - t]'/t * zr - 12' (' - rY i * zr - 12
lr + z, - ? = z - (' - n'll
-$-: - 1
(, -t)2
&
'd' r'd' t'f.,Fll=-#4*.=-'m=-a'v-.=-$.,
t = r
-4-=1,- r=1 =,= r * l=*= -1r,1 ="t'-,PJ6fu-, 1 Y Y Y'l
-la= t $, = -t ffi= -it'l" -'li 05
" -'1' -il;li
v
r=tr+Iz=-;"8.
u,.t$VN
\=l
HaIIL Ovvalcs kerr iizoritrdo sade gevrms aparaq:
,=,q+*=vfu=*,;vfOnda verilen iroqral iki iteqral& c.mindon ibaral olur:
, = tfu = t #::*. t g,t'p:. r=,,],. f ' .il * a,
.&t,=tViW=
=t=x2 =t2t2 +2!z - tzfi - P)=a2 -r2 =
Jrz*z-!'lx'+2
x2 +l6=ar=---? =41
lxz +|'li +z
.&=l-=l'V2 *tN,z *z '
l-a:l- l'-'-
12 +lij
=1 d'.=rnrg=*"rg-+:
l+t' ,lx'+2Belslikle, bunlafl I idcqralmda nazsra alsa4, yazanq:
, = t$; * = ^!y
. l?il. *",e f,r, * "
Kvodtat fr+hadlinr kmtonik SeHo gatirin, inteqmh topn
Et* [-=--$-=: apn.'(r2*r*l!r2+r*tHalE
2t2
ml
x
2r2 2ir2 *z=! . rz=-!-
I -t' l- t'
r=r---E-=, '('..r) =-=. (,2 *,rr!,2 *,_, .l[..
i),. i](,. ;), _ ;
=L* l=, =, *=a,l=t-- d'--.-f 'z' - * *1 ,(,r,;),iF-
nuoA" f3 =r evazlemesini apara4:
1'I1=!;4--= t=1nl{:}4
2' 3 ,,2 .16 | ,13 _2y2B- )/
Buradan kdhns dalpno q&yrtsaq, alanq:
,=l=JfJe l./r -(zx+ ri)^li
w.t_L:' | -2, *
'2l,lz * z, -,2
Eatll lfreqtalafu, kesr iizcrinda muoyyen sadelagdirme aparmq,verilen imerqal iki irteqralu cemindan ibarat olar:
. ,2d, &'' (+- z***'lz*x- ?',iii-l
* ;__?Jl!:_+ = t y * r r,'Q-z'*,2lJz*z'-,2
. dx db-l\ .r-l=r--:: = g:aEm
-'
"lz*2r',2 'rlr-(r-t)2 J3
Ir -imeqralma r -l = z evezbmesini tetbiq edak. Onda 1, inteqrah
aqa$dakr kimi olar:
2z -2 2znz ^, ,lzlc = l--.==--=42,'' = t
F;?Y77 = t 6;5, -7
-'t TeE= = t i - 2 Ii'
+c.
n2
1-), lJ;:,; =,, ---z--^-42=41. It,--t-'-=--, -,: ij- z' i='' 'P*r2l,lt-"21",=t-,r, t+z2 =r* j_12 =6_r2l,
=-t -4-=-):n1 ,.4 l=-'=rlE ql'6-12 J6 l-t+J6-zr J6 l-Jl-12 *rfl
_t dz -,_ 3-"'.,:h=' 2 = 1
F;7fi17 -' fi.$-y''[.j: -
i-=:=u. '2 ==v2=r2=4.lJ;,' '' t-22'- t+r2'-lr-"'=
'=. t*12=3'6tI 1*y' t+y'
3
lr'*"ror=!J ! ,413 -=Iti *oyz-' - 3'1r2r2 6' !*rz
I+ v'11 ,^ .11 Jl,'- -:d.clgi:lt =--i"'"'""'-O*'"$_j'
r. = t \ - 2r':,= -lhh!--4!q | - !1,.,r EJe -l-1t-,2 *"61 3 " Jt-r'
,I, irtelraLnda kohne deyigane qayrtsaq, alanq:
. r.l.[2.2,;-161 5 ..D(x-r)12 c -_LlnL+=+::=-: l-
=- arctS'-r:=--:
^ + c '{o l-{2+2x-rr +16l ' ^12+2x-x'
{ va /, -nin tapftnts qiyzretlerini verilon / inteqralurda nsarc alaq.
r-l r -l liz--,'-AlI =l\ +11=arcsm---lnl :.: l-J3 J6 ]_,/z+z,_12 rJo,
,l /ji,-t)-
=-@crg -=-rc.
r ti2+ 2x - r"
v
mI
SM. r=d+fr bsr-xetti evezlamasininI +'
I
@:.W _, _ tirreqral-, tap,o.
EellL
,z -rrr=(orAt)2 -a+fr *r-(a+frY -lr+t\d-+ fr)+(t+rY -(t+rf t+t (r+rfa2 +2aft + P2t2 -a-ff-a-fi2 +L+zt+t2=-_--@
,r. *r*r=@*P! *a+fr*r-(a+frf + lt+ t)(d-+ frl+Q+t)2 -(t+tf t+t (l + t)'
_q2 +zofr+ 02t2 +a+ fr +a+ fr2 +l+2r+l(r+tf
Bu ifidalordo t-nin omsallarnr srfra beraber etmekle a ve p -n t"I*tq:
[up-a-p+z=0,-lT:1=''l2ap+a+p+2=o l" =r, p=_,
lnnlari sy5l6modo nozoro alaq. Orda , =
= o6r. 6 = :!, , =l-' .- -- '- l+, --- -' (r+rf ' - I*r'
,'. ,,-.l=3'2 *l=, ft; =(r+tl
Bttll bunlan irfieqralda nszors alsaq, alanq:
' = q dffii = -4 Vfu,- 2t Fffi = \ + r,
kdmsyi ils
,lr';1+t
Owelce I, inteqralmt tapaq: ,r=-rl@fu, n raau J2 * I = y
evezbmssioi aparaq . 7f1=0r,,'=t2
-r ol&gun&n
ttz +r=z(yz -t)+t=zy2 -e
I r .lzJr-.,[r,'-:llI z,le l2g - ,lyt2 +t1l
olar. Bu halda
11--21-!- =-21-9-- r ,S*+' 3yz -8 ' a- 3yt 2J6 12,12 - ,l3Y
x1
1, da kdhne dayqana qayrdaq. Onda alanq:
indi ise 1, futeqralmr tapaq:
12 = -2!7 ^- dt
p,";F1, futeCralmr hesablanraq W" +=--, avezLomasmi aparaq. Burada
,lt'+3brmdan owslki ideqralda oldulu kimi harakat etsak, 1, inteqralut
aqagdak kimi tapanq:dt -. & l, &
r z = -zl g,z {,2 a=
-rl Br\ t= - it,\ !=
'- - J, z,fa= -lzJLtup2J2z = - ,*,8m.
1, inteqralmda ko'hne dalqeno qaytsaq:
.[z zJi\ - xt .[z "!1Q
- *'tI 2 = -
-@ctg ----i-- = - ;atcrg -I--: .
' 2',lr'+x+l ' !x'+x+lBelelikle, verilsn futeqral ag$rdalo kimi olur:
I = h + I. =--lt{oJ, Oo -,r'-0c'8--:1+c' Vl-+r+l
Eyler svozlemelarinin kdmsl ita agalrdakr imeqrallan tapm:
tl Jd *tt*" =x.for+r, a>o;
zl JJit"=o+16, c>o;
:l Jar=x;-;r--ft-q)sLt.l--+:
r+Vx'+:+lHalll Burab a:l>O.
edek:
(1,,=+t]vol+rW,$d."-p
^ll-,.t
itrtEqralm tapm.
Odur ki, Eylerin l<i evezlomesindon istifids
m5
x2 +x+l=-x+2,y2 1a11= x2 -2p + 22,
,(+z)=72 -1 - -=#,, 222 +22+ 2
dx= -____:_:--:&.fi +Zzf
Bunlan verilon ireqralda nezsrs alaq:
- -222 +22 +2I =l- -:'--+:e4t + 2z)'
Id€qralah firnksilanr sada ksrlon ayuaq:
222 +22+2 -At B * C -ah+qz+qz2)+*+Zz2)+Cz
4 *zrY z t+22 (t+zzf fi+2zf$.t +za =2. IA=2,
ltt* a *c =2,-]o = -t.le=z [c=-r
Onda imeqal a5agrdatr kimi aprhr:
, -r2zz +22 +:2*
-zt& -y & -lt tu . =' 4t+2zY 'z 't+22 '(t+zzy
= ztng - 3 nlt * z, + 1. I *"." 2' ' 21+22
Bwad^ , =t+G\ x+\.Et2. I =!
&t;ffi
EaUL c=l>0 oldugundan 2+i Eylcr. evedemesindon i*ifrde etuok
slveriglidir:
.llij=,,-tl- 2x- x2 =a2t2 -2y1q1,
-2x-x2 =x2t2 -2xt + -2-x= 2-zt - re+t21=y-2 -2r-2 2+4r -2t2 t2 -2r -tar= -----; - at= ----------;--;-.1t, E,-l=- .-.l+ r' O + t')" 1+ t"
Bunlan verilan ir*eqralda nszsrs alsaq, alanq:
z+4 -?l *,=W=2ffifialffikrt---
l+fBu iateqat rrda krrho sytrEs ts{u ilc tcl oddc
t+b-F / B Q+D(r-r)fl+l; t't-l' f a1
_,{P -P +t-r)+N3 +t\+c_(!3 -?l+o(P -D.(r-txt+l)
[t+ t+c =0, [r=-\ [r=-r, f=-Ll-A-C+D--L lB+C-1. lX+D=-1, la=l{ ={ t{ ={lA+B-D-\ l-c+D=4 lB-D=t lc=o[- r-t [8-D=3 [c=l-l lD=-z.
Onsdttm h dyn*ini t htcqrh rynlqo0a ncac rhq:
r+ 1
Tprlm5 h qiyffii wih tagalda me rh4:_ I _k2_ 4tI=J1,-r)i,-11,*r1'atBunda sr& tsbra ayrnado i*iffdc od&
r,-0" ,='*#E7t
ttr' | = Jiffia'.W. x2 - 3x + 2 = (x.:1)(: + 2) oLfoSe irEqrelts
qrhmde Eylcrin:<'0 cvcdmcdndm iilifidc c&k.'r2+3x+2=t(x+1)=(1 -r- 1)(r + Z) : 11** 1); = x+ 2 = 12(1+ t) =+
x+?=xtl-tt2=r(1 -t2)= t2 -?..2- tz
, = - ff . I*-, ffi= -W,. * -4-z-,as' *, = fil -2o@t + c
-ztdtx = F==+ dx = 1,: _ 11:'
"x7+3r+2
-Ztz - 4t A B C D E
G:n[ll11a13 (t+1)3 -(r*1;,' r+ 1' r -r' t -2'
-2t' - 4t = A(t2 - 3t + z) + B(t' - ?tz - t + ?) ++c(t4-t3-3t:+t+Z)r-
+D(r+ + 13 - 3r2 - st - z) + E(tr +2t3 - ?t - l),t=l ysrsaq D=i,t=_ryaaaq n=i,
t =? ytry E = -';.Soors t'r v5 t 3-04 mslllcfl m&rylse Gda&:
I C+D+E=O, ( 16
(B -c- D't-?E =o=tc* D = -= c =t7 33? 5
t7108
B=C_D_2E=_106_4+n=G.Oms.lhna hr qiymad*ti ayrlSda maa dsq, / itrSdI etq&drtr
tioiob:rf dt 5 t dt 77 t dt 3f dt 16[ dt
I = 5 J G +lF + re J rr+r): - roe J t + t- i I t - r- n ) t - z=
1 I 5 I 17 3 16.
,V'-Ti + 2t =
-.
Brrtda .r + ILfutilisttllo t*biq ataHa btqmllan topm:
fri- t= f--+:.' lI-I'r\'l- t_
Eefl, i= t'z ovubmcsini ryeraq:t2-1.
I + x = t2 - t2x= r(l + tzl = t2 - I +.r = pj- dx
4tdt(t2 + 1)z-
(r, - r)3 112 ;- 1)3 - (t2 _ 1)31-13 = 1-G{lF= (t2 + 1)3
t6+ 3t{+3t2 * 1- t5 + 3t{ - 3t2 + 1 6t4+2
. (t: - l):l-(,=1-G.TIF=(12 a 1)3 (t
tr+zt2+l-t*+2t:-1(r: + 1)2
z + 1)3'4t2
= Gr+lFBu evafuel*i inoqral& Dffie d4:
t2-lt= I - iFTf'G--+D"- , ZtTn-'Zt-:
=li#"=l(;;#)"t ar dt
'=l-;l;; (l)I j-L irt"c*f- fipmac 090n t6t = y avazlanesini aparaq:
a, = .r=ar,VJ
I dt tf dyJ3t1 +7 \t1 )v',+r'
Axrmcr inteqral sada kasrlere eyrma 0sulu ile 452-ci tapSrqdsalnmldr. Hemin naticadan istifedo etseh alarq:
I I dy 1[ I ly2 +,12y + ttiI t' +t {11+.t1"'ly2 -,/'2y+1
Owolki Y = V3t daYitsnina qayrdaq:l.***#l
1 t dy 1[ I lv3t'?*trzt+rl r t,J3-11fi I y, + t= 16l4,tztnl,r1r.- - t51u 1l- -r"'un -.,E-l'Almmrg bu naticeni I inteqratnm (1) ifadesinda yerine yazaq:
(1- r3)v/r -7
x2 -l
Grll)5';7;7
r 4llr' /=3- tfilnvzt"Bursda t= /t:.
'{ r-'
^tir2+',tT|l+t_-::----_-_':_--J3rz-v12r+l
t cr./3 - rl- -arctn
--':-r'
EtL. bl,=1,-' dr.- (rt +l r\xt+l
I7 rd, Inte$atalh ifadanin suret vo mexrecini :z-na bdlak:
r-1x.fut= fut=(x'?+ r)fir+ r
a(, oG -+)G;5ffir xz_L I x,lz
, = j 1_, ;r75da = - \liarcsin;;;l
+ c'
erntffi.frZt Intlfatn kesrin tlzerindc 9ev*" *' ,,
------ - _.----:-xJT+ 2x2 - l -l 2 1rrJl+F-F x3
Bu ifadani in@ratda nezre alaq:
t dx L{'(t-*)l, =rrm , t-l=-arcsin-l?+c=
! xz-l= - arcsin----=-=- + c-2 JZx'
I x^(a + bf)pd:r binomial dfaensialu inteqrallatonast:
fi,*a" ,, z va prasional od.dltrdir' Bu tip intqralm t4lhastasaErdakr 0c haldan birina eBtirilir.- 7na, ptam adoddh. Onda x = zN avezhrcsini +armaq lazmdn,
burada IV adedi z ve z kesrlerinin or@ mexrecitlh'
17 16, t! kesri t"m adaddh. Onds a + bx" = zx Ourads i\' adedi
pnin moxracidh) evadamasini aparmaq lazrmdtr'
tII hal: f*n tn "i tpm adeddh. Bu haHa ar-n + b =zN
(buda' iI adedi pnin mextpidir) evazlanxiei apcmaq
lazrmdr.
AlaEldah inteqmllur, taPfi :
tt7. I -- ll=da.( 1+ 1''
210
-2
),-(r-*
Eotll m=:, ,=;,r= -Z p = *2 tam adeddir. I hah tstbiq
edak r = t6 avedarnesini atrrra{, dx = 5tsdt-Bunlafl verilen inteqraldanozore alaq:
, - i## =, I F#n- u I (,. - zr, +s - vff]r)at --!r'- -+13 * ra,- [p]t'ia, =
=!1s 4 4gr + 1Br - 18 I d'= - 6I t't=
=s .l+r. . (l+cz)z
= !1s - 4r: + lat - 18arctr, -6t #.I t2dt I t=u', dt - du; I
It =J i6ry= la,.-]rr +r!)-rd(l+ r,I r- -#;Il -
t tl. dr r l'= __ 4 _ I= -ilil;t *r) L+t,=- 21:174*7*"tst
Axrmcr ihdeni (llda nezere alaq. Onda alanq:63tI = ;rt 4r3 + 18r - t,ia'ctgt * 6;13 - 3arctgt =
6-3r- St' - 4t' + LBt + irlJiJ - zlorctst + c.
Burada t = 6v?.
sssl=!L.nr+17
I
flattlt = 1 r(r + ri)a ar. m = L,'=i., = -i.
= = ? =, - m ededdir. Bu hilda trhaLtetbiq etmek lazmdr:
1 *ri = t.2 + al = 1z - I + x2 = (6r- 1)r "eZxdx = 3,2t(tz - l)2 & -
xdx=3t(t2-t)zdt.Bunhn rnleqalda oazare alaq:
,= i,0*,i)- a* --
(1)
tJ'ru-rr'at=tt'
,(g- 3,'+t)+c
(t'-ztz-r) dt=
2tL
,r,Buradat =.'J 1+rr.
W. r = !'vTr-:Far.flallL 11r1t = I3,tTt:Tav = -f xi(3 - xr)idr, ^=.,n=2, p=;
* *, =t - i = i + I = r u- odaddir, Bu halda III hal: tatbiq etnek
laamdu, Ysni, I -3x2 - ! = t3 + 6xdt = 3t2dt -- xd)( = -tzdt.
Bunlan verilsn ideqralda nezsrs alaq:
G -il.';n*=il"(i*l=I =t,du=dt, I
:l t L \ I l=ldu=,1(F;lJ,r=F*rl
3 t -1,:i (1)=Z'tr+t Z.tt!+lf I irr*.at rrn helli verilib. Odur ki, hcmin naticaden i$ifada edok:'t'+1
f dt 'vt+-I 1 Zt-Llffi= t'ffir- JTnrctsli-''
Axnmor ifrdoni (l)da nezors alsaq:
3 r 3l Vt+I I 2t-1\r = 1. ,a t- 1[rr ,,561 - fiarcts -i- )
+ t.
Buradat=',11i--t.ASqEtfuh l$eqrullan taPm ;
E9O ! sina xcoss xdx
^Eallt Sada Crvimo aparzrq;
I s.no tcosrxdr = [ sirr. roas.rd(sirn ) = f sin{ r(1 - si,r'x)2rl(srnr),-**--- J-- r
f= J
stnar(r -Zsin2x + sin{x) d(sinx) =
I= J
tstno: - 2stttbx.- sinsr)d(sin-r) =
sirrs.r - sittT x stne z: S -Z ?-+ 9 +c.
8n. | "t"l' dr.'coft
Hemf sinsx f sitt2x ll-coszxJ "."rrb: J'---';sirudz = - I *l;d(cosi) --
. f d(cosz) t d(cosx) 1 1-_ I _-t I __-___-____: __ _-! -J cos't ' J cos2x 3cos3r cosz' -'
ssz t4.' sl:ft',HeA
I ib. f coszx+ sinzx I dr. f coszx t xt
I ;Fr= 1 ---;i*i-e - J "r,,,*J #;dr - tnltsd+ t (r)
I coszx 1.o", = r, du = srn-3:d(sinr)lI = I ...-..-.=-- dt = | 1 l=I sinx l-sinxfx = du, t = - Z"i"\l
dosr I sbtxd,r oosx lf tlx cosx 1 t rr= -Mi- I:,,'r;= -:*'FI-1J"i, , = -2"h8 -ik!szl'Aronno nefieni (1) - do nezare ahq:
ftlxtxtcosxlrttJ Ar; = tnlto1l- 2r;f ,-Zrnltsil+c
1 r xt cosx=Zklts zl-.o;7;+ c.
t$. b I s\-!.--+cott z.fr[ KNir suret va mxrecini cos8x-c b0bk- Onda almq:
i= 1a*g:o' ixtsr) = I!!4##@ou*,= | tg''rd(tsi +t I ts-',a(tci +t I dhsx) +
t131+
Jtszxa(tsr) = -id;-;.3tsx+;ts3x+ c.
t91. lWsxdr..IIelll
1 = [ tsrxdt = [** = - [(1- coszx)z
d(cosr) =J J COS'X J COS'X
- -f | -Zcos2t + cos'x
d(cosr) = - lcos-sxt(cosi+J cossx
+zIcos-'xd(cost)- I+P11: '---:- - ----=- - Inlcas:l + c.4edsla cos'z
ses.f &"- J*L',Eo'.,Ea@ Kesrir $trat ve memcini cos't+ bOlak Onda almq:
1 : 1*Jrqa<,,s') - lo *'{i a6s,) =' t ,fn J ,[tF7 --" '= [ toi a(tox) + | tsi xa(tg,) = -2rsi, *zers?, *
".fr,6. le--- lagzE ?fl.Btizdr^ tg, = t2 eve?Tfresindan istifrda edek Onda
x=.,rctstz -ar=ffuat.Brmlan ia@ralda nazare alaq:
2t
'=l*'o'='l#"r h2 +At+il t t.lZI = ?"{ztnlv - rrzt + tl+,,liatcts l- + c =
L .rfiai*-=atCto-*C..tZ - l-tgx
I sh.astrls = f,lcos(a - F) - cos(a + 0)ln coso.,cos[ =f,lcos(a- $ +cos(a+ F)l
rrr si,,t,aIcosS =f,Ist'.(c - il + sh (a + F)ldo.cturlornn letmayila inleqrallot uprtt :
tt 7. I= I st:arsir.isi'littt.IIa[t Yrsandakr I <t0sultdan istifade cd.k:
lxxfx3xxt = J sinrsin rsnl* = J Gos1- cos-r)siz5& =
214
r . ltsx+ EEiT +t:
-ln}----------z,l7-'- ltgr - JTdi + t
llt x 5x 7x 11r\=;J (-stz;+sin7+srn7- stn , )dz=3x35r37z3Ltx
= 7cose- -ocosT-frcosT + icosz-+ c.
Egt l: t sintzxcoszSxdtc.EaIII Yuxrdalo Itr dlsEtdan istiftde ed.h
r = | stnszrcosz3xdz =| /fr"*, - "*rl(1 + cos5z) dr =lt
= , J @sitOa + lsir8xcos62. - sinsx + silt6xcos6x) dx =
= i/ (r",., - lsioax +lrsaax- saer - ]sarr4* =
= f,(-1"*r, *1-"+, - * -"a. +!"*"* + )*":nr) + c.
I sta,(a- F) = srz[(*+ c) - (r + PIIL cos(a - fi) = cos[(r+ c) - (r+ f)l
eytriliHarinin kinayila hteqrallot tqu:srP. JffiL.x.t *rl.Ee,A
t d.r,:1-' J sin (, + a)ti.n'-J sin(rt a)siu (r + b) sin(a-b)J sir(z + c) sir(r + D)1 fsin(r + a)cos(r+D) - cosk+a)snG + D)=.i"("-D)r@dt
I flsiuc+c)-(z+D)lsin(a - b) J sir(z + c) sir(r +D)
sin (a - D) sin(z+c)sin(:+b)1 I cos(x+ b) 1 f cos(t+ s) -
= d" (. - D) J ;;G;Dd' - si(" - D)J snG ;1 ar =
= Aal-,Ialsh(x + D)l - brlsin(r + oXI + e =r - lsiu(: + b)l
=,;o 1"-61h1*6;61+c'gel. I e.
- rhr-d! 4flcu
sinz-sina: z"*ff."ff,","o='o"[ff)-(=I, )
okfuluadan
IJypan triqonomettlk a'azbmal*ln fumayl ila inteqrallor tapn:
sot. takfrdtt Bwadin tsi = t universal avezlamosindan istifade edek:
x-zdti= *nn, + dx = 7l{z
cosx va sdnz fimkeiyElarmr tg i frrtrksiyor ile ifade edak:
. zsulcasi ,rni 2tsi,tx = ;,* s#z= t; t s,tr= TT*'
cos'*-sa'4 t-w'4i t-t'cosl = -# = ------i
-- :":=^x,cos';T;*i- r+$'l L+tL
Bu qiyostliri ve evezlemeni verilen inteqralda oezors nlaq:
' Zdt, - l TTtt-'-l 4t L-tz.--
T+1;it-TT7z+-'' 7' d.t-r l
--
--tfiz+qt++
-. I d'--J*-t+t2+5+stz
216
,- ! r-"[(#)-(+s)],-:'=u"*J-Wry*=, , cos# co"L# + rio'i,o
"in'1"o- ]
-n'=
- Zcosa! snffcos--7-r x-a x+a Ia I r cos--- f sul -=- |=v",-v#*.1;+*l=
x+a.1
= -lncosa
+c.
tt dt r1a(t+$) 1 3r+1= i I ;44=5J (, - .9=
= ltsscts -f,T + c'
1 3 tg! +t= Garctc--G-+ c.
go2. I= I 'inooo dx..'dnt.+.orz
^grdl Kfirin surcti tzcrinde sade gevirma apraq:
sinucosx= ]Ct , * "ori" -f,.Bu yazlql in@ralda nezare alaq:
r -i I o v*':.:r*- t
a, - | [ <, i,* + **) d- -il #,*=l,.,rn -"o"r)-11.2' ?' (1)
I da t x 2dt r
"= J "rr*
* "*r= lccl= t ar=r*trl =Ut
t ffi ^f & ^f d(r-l)= l-Z;-.=A=') r+2t- tz--1J z-1t=fr=
TTltaTqV
Axrmcr naticcni (l){e nezera alaq:
1 1 bc*-r+alI =r(siltt_cos:) _ mr"lffr_rql*"
goi. I:[ tu",'-"ot' dt.' th'x+to?z
EaIIt Kasrin suret ve moxBcini cos'r-e b0lsalc, onda alanq:
t = I'##dftsi = ttgx= c irrcedard = I #" --
= ['-* ,, - t at'*ll 1=)4*=Jn#-=;JikH+#1..
t lt-t+fr=.:lnl------.-..=.12 lt - 1- ,12
t b2 _ Ot + tl ! - lts2x -,|-Ztsx + ll= ffi h
lV +dnl+ c - ?fr'n F;; *,J-r*, * rl+ c'
cu. F ITT dr.- l+slnr,EalL Sfu geirnalardan sora alarq:
, = 1 ffia,= I#3 = lsinx = t tera edekt =
= I # =if ffi =!**e* * "
='r*nsl"iazx) + c.
n*t Ie#a.E flt Ke€rio $ret ve qe:crecini cos5:'e b0b&. Onda ahq:
, = I #er= I 9:##! = ftsx = t is''a dokl
f (l + t2)2.1t [ (l + tz)z ). -= l--An =J [,*q1r._rnr".-=1ffi,=l*,,=l-+-
= *,re(-i). "
= -no(w* -#)., =
-,agg#+c''e6
Jlffi itz = Ax +Btnlosi,.z+bcosxl+ c (a'B'c
eebithrdh) oldu$mt tsbd din'hb.E
ars&rr +hcosz = A(or;ttrx + bcosz) + B(a.osx - D"qI) O)ffi. Trirlctrcnfiil fimksiyalrm cmsallaru m0qayise edak:
{f =#;'}'li -!l: a2 + b2 * o'
A= @La + hb
'-dr+rz',=*l-??.- a2+b2'
Brmkt futsqalda yerina yaxaq, berabdiyio dopulu$Du alrtq'
qz r=tglg-da.' t,', +2.coEt
.E#, hEqalm @rkoeqmd! €walki hsttda istifrda ed.k.aa= l, 4,=-1, a:1. b=2.
I = Aa, +Blnlos;n + Dcosrl +c (l). aaa + 4b 7-2 1,:-=" az+P 1+4 5'
ob, -ba1 -7-2 3D = ___j_- a2+b2 l+4 5'
l, A -nin qtynetbrini (l ) dlbtrtoda yerina yrzsaq, idcqal, trymrg olrq:t3I = -:--:tnlsinr+ Zcosxl + c.5- 5
,r& f rrr*.trco'alcr dt = Az + Bhtlosi'u + lr,osr + cl +'c I &- -- r .dE+Lar+. , .iE+L.E a.
(f,4C sab,idadir) oldulunu isbd odiahw.
oi,sil'4 + \cosr + cr: A(o6iru+ bcosr+ c) + B(d@sr- bsi7.al + c (1)tini gOtOrek Triqmometik fiEbiydrim smsallrm Euqrybe etselq
almq:(a-Bb=a,lAh+Ba=b\(Ic+C=cr.
Xati tonliklsr sisEmmiA,B,C-ye gpra hell esak, alanq:
1 ="'?'+ !\b 'a2+b2'
B=+-!?.a2 +b2 '
6=q_Ac.Omsallrrn @dmA qiymerini (l) b*abnliyiod. yaa4 isba aydmok.
9D. I tu+2cos:2 dr.
' A.r-2c68+tEclL Blrrde,l2S n0nreii @gmqde istifrde Gdak:
at=1, 4=2, q= -3,o=l,b:-2,c=3,. d|a+\b l-1 3
! =_=" a2 +E l+1 5'ab, - bq 2+2 4
P = ----:-- a2+b2 5 5'3 9 -15+9 6C=ct-Ac =-3+r3--3+r= S =-S.
Ish'l+z@sx-3I
-ir:
J sa", - z@sx + 3-' 3 4 6f da= -;r +;Izlsru -?rosa + 3l -EJ st'-_Zcosr+3'
bdi I = r,r,,-+=- inteqralm t4q. Bumda tsi= t, da = #, ,
- 2t L-tz ,sinx = ** cosx = ft qiymotlerini inteqralda oezera alaq:
,_[ d, :t iear- _' - ) ant - z"osx + 3- J T?i -);p *,-"[
dt =rf d' =-'J zt-z+ztz +3 +3tz - J
'tz +2t+ !
t dt 5t + 1 5te;+1= Z |
-
=nctq --
+ c = daS --i- + c.r <,/Et+;irlr+i
dknrnrg netiomi yuxanda nezara alsaq, yazrq:f sifi + zcosr - 3t-I sbu-Zcosx+3
5 stgf;+r= - 1, * I ml"irr - zcosr + 3l -l arts":!2::: + c.
3+5-'5 5 2
910. ! x1 e-* d.x.
Eelll
[ ,, "-* a, - lx2 = t + zxdz = atl =lf *"-" at
I t3 = u, 3t2dt = du, | -= ldv=e-rdt, v=-e-rl-=}?uu'+t I t'e*at\ =l;;;:'*, l;=-:;'-,1=
=!l-*"-, +3(-t2s-t +z I te* atl]=
= -'Zrr"u -3rtre- + r (-te-t + | e* at) =
= -Lprn -f,tre- - 3te-t - 3e-. + c =
220
= -e-t(+.T.sr'+s)+c9tI. I = J xe'siazdx.
Eald, Bu haHa ,rro = t{ tar"iodm istifsde ehek hamdr:
t = | x e'sinxb : f,[ * "'{"" - "-i')at =
: ll, "u*, a, - f,f, "o-,'a, = h - Iz,
t, = 1 J'
*,,n,o " =l; :;; ::"*';:,"1== rl[rf ,,,',o -*1,<r+0,2:l =
= iffi .u.,,. - .,fu eo*o'l = jffi(, - *),, = *1,""-'l)x tr =l*'=l i== !'J::,r,1=
=*l*"u_o._nh,,",r1Aluun axrmct neticaleri I = l, - Ir-da nezara alsaq' alanq:
,:,1ffi #-fr"r,n>.#],rdxtdtr'=J rntr+il' "=J t(1-,)'
9tz. ! cos2,ti ik.HalL r=t2 avazlamesini sparsaq, dt= ztdt. Ovezlameni
inteqralds nazara slsq:
r = [, ,or,t .rral- = t {t + coszt)dt =
= l, a, * | t cosztdt =f, * ! tr*zrtr- =
_lt =u,du = cosztdtl tz 1 1f= l* = ar" = ir*rl= v + rtsi,lzt - 1J
suattt =
t2 1 1 x t- - 1= i + i$rn2t + ;coszt + c = 1+ 1.1-x
sinz.ti + Tcosz,li
+ c.
en. !fia,.trI Zf ksri &zg0o kesra gevirek:
,= I#*= I@-,*,*,1-)*=t dr f e'dx=er-z+I-=ex-r+l-=J t+e' J et(L+e\
= "" - * * I (* - #),,ot = e, - t., - [ +-i* =
=e,-lnll+erl+c.glt { +?---vr+.!+.r+.-t
A=1, B=-=,z
Bu qilmctbri apl$da nezere alaq:
Onda r=6tnt oHugundan
A= l,B-D=O,
B+D=-1
1
2
ECIL ei = t rvezbmeeini aparaq.r alru r 6lrrt
si = aT = f,sl - s-T- = g2
olr: di = ! da Emlsr inlaqralda uezare ole.l:
-fdzfdtfdt' = J ;GGG= 6J mr rfi4F, = 6J ;ir + fuo.
1 A B Ct+Dt(1+tX1+tz) t' t+t' L+t2
,t(1 + t + t2 + tt) + E(t + t3) + C(tz + t3) + O(t + t2)r(1 +
(A+ o +c =0, ()A+c+D=0,-JBlA+B+D=0,-lc\ .tt=l \B
1
tX1 + t2)
A=t,+C=-1,.+ D -- -1, '
+D=-L1
D = --.z'
':"U+-{*.r#"1:: e [rnttl -'rtnt, * a -lrnt, * fl -l*nsrl* " =
= slnl"il-)nl* "21-!rml
*,il-)*no"i|*. =
= off-lnl, * "i1-!nnl * "il*t *ao"i] * '== x - shll *,i1-|nlr +,il- 3.'.asei + c.
,r*tffi.ECfr. er = t svademasini rycaq. ODds r = lnt oldu8uod"n
a, = iat Bunlan hreqralda nezere alq:
, = [ E i,, = I #- | #= ah+ li' -it- t,r dt r 1 1. I
t,= J ,,,VL=lt =-, dt= -Ad"l=
,_#o" f itu .L= lT- -.I,cstnrt -- -arcstni.- I"-:a - Iuru'
Br4u yurnde f inteqratmn almmg ifidainde nozere alsaq:
I = hlt + JT, --4 + -rrrnl + r.
Kohne tleyigma qaysaq, alanq:
, = I ffi* = ,nle'+ G;-1+ arcsine-r +c.
e16. t=l*:- Vl+._+i,l-?'E A Kcmifr suret ve Eexrecini mera'acin qogrnasms vuraq. Onda
almq:
,=[ I r JTAir t r,l't-e,tu=ZJ ,, d,-il * tu
223
.ITTA - \6=7
11= it, - rtr,
,, = | ff * = l::;;, :i,"*==, u;i 11
=
t tzdt lt = u,av:)ru - r)-'a(t'- r)l _='11v=o=l dt=du.o=-rG;4
|
r t lt dt 1=zl-*:i+2J ,' 'i=r I rt-1r .'lTTir l. |'/IT;E-11
= - t, _ t* rt, l, * ,l -- - ,. + ,t,17n-r" " 11,
. {'fi-r, 11-e'=t:="-trriil=t,= J - r, tu=
lr= r"tr - i),dx=;+l=r t2dt lt =u.ar= -ia- t':)-'za(r -t':)l=-'lc-+=l o,=ol.u=#a l=
--'[rr='l-;l*l==-*-;'l.l =ry i'l#l
Bu naticalsri ,I-rrin ifadesinde nezan alaq:
1 1 ,IT+T 1. l/iT;'-11t =1tr-ltz= -1;;-+',, 1u1=-r,*rl*
,tT= e, f- lrif=e'+f I+- --ln l-=-:--l *r.' lex + l-Vf - e'+ 1l, .r2
et7.t=[\r-:)e'dx.IIaIII owelca sadelegdiralq soffa isehisse-hisse inteqrallayaq:
, = l(, -!,. *)"* = ,' - rl";* * n l"-* =
224
= e' - 4l{& + 411
Bu ifudai (l)da yeine yezaq:
. t = e, -+ !Ia, - ql+ q Ill dx = e, -I+c.91s1= 1-!-da.- xz-3u+2
Ezfil Moneci vuruqlra ayraq:
, - Ic=#4*= I"*(*-,-,)*=f e2' r e2'
= l: ^a,-I ------: tlx = st y;("zt,-z)) - "z
1i7"2<*-t)) + c.I r-2 J x-7gt9. I= J lnl(x + a)*"(x + b)'+blG#crutPaId, lnrcqrataltr fintsiyuu bh qeder sadalagdink. Onda alrq:
/ -= /f':'#). '(t-2ld' =
= J[ln(: + a)d0n(: + b)) + ln(x + D) d(ln(r + a))Ff
= ld{ln(x +a)ln(r+b)l = ln(: + a)tn(x + b) + c.
e2o. I tn2(x + ,{t +V\ az.Eatrfl, Hiss+hisse incqreltema dmrunu tstbfoi cdek:_ u=m,G+Jr+_r),
au = 2h G + {-) L,tu,,. (r . ft) * =1
= zta (r + tll + x, ).f,iradx,la=il1s+e=a
Bunlm verilm ineqralda nezen alaq:
ln'('*Jt-r,1*== xh, (t +.{-+ ) - z I nQ * \fTT7). fi* =
(l)
225
: xbP (r +.lTT?) - z l.l taV ut(x +,tT +V) - l .tTt?' # *l-=,h,,(, +JT+;,)-zl,tTTFn(, + Jt+i)- xl
+c'czr. r = J tn(.lGa +.lG-) axEaEl Hiss+hfuse itrqdlm! d0shnmrtstbiq edak:
2= 6({f:l+fi11)du=G+Gr?i=.il*)d'=
LIT+x-lT- dz
2.ll-x+,lI+x ll-rzdo=d'r, o=x.
Bunlm verrilrn inEqralda nezae alaq: .t?i{t-,n- xdz--:,=,t|-xz,tT-+,llTr
u= Ir(: * Jur)t fiT7+x. d,au=;;.,tr;;,_'11+xi az
zdx Lf- I
-do = J#;- o = ,J{t+ xz)-d(r + x2) = ,tt+ rz
r =xt,.(,lT=+rr@-:lt r1-lr= -
= xtn(iT - z+,tt+x) +VJ,ti: da =1:
= : Iz(V1 -.r+Vl +z)+i*csiax-V+c.e22.1=- ! (th|da.
. EaflL Hissahissa incqallma qaydasm tr6! edak:1
t=lnxsdu=-dx,xd.xfdtt
da= r*ffi-v= J n;;ryi;= Jr,t'(r +$)%
l\'12 t 1\ x
d,
=-ilu
Bulan heqlalda nazere ala4:xlnr I dx xlnt
r =JT*- J I,l l.r + v I +.rzl r- c.
923. I = I xarcsin(L - x) ilz,fa@ heqrahlt ifida thinde m0eyyfl govirme ryarag:
rcrllln (l - r)dr - (1 - r)arcsiz (1 - rH(1 - r) - arcsin (1 - r)d(l - r).Bu ifiddi intemqaHa nszara alrq:
1= [ rrrcan(t-iA=)lf
=r(r- x)arcsin(l -r)d(1 -r)-J arcsda(1 -z)d.(t-,) =
= 11 -r = t ib islarra eitakl= t tra.csin ttt - I ocsin tdt = L - t z,. lu= arcsint, dv = ttttl
t,=Jtucsintdr=l * =#,"=: l=t2 I t tzdt--V6csittt-1Jm=I t=u, ttt= du I
=l* = # -, = -)l o -,t,)iao- d) = -ft - r,l =tz lt f
-
\=-@cstnt -I(-tVt -tz + JtlL-t,e)=
=f, **,* *lt,tt - * +l[ <t - t"1i 411 - s,1 =
='j,*,,* * lt,{ r - * * lo - *>i. lu = arcsint, ttv = dtl
r.= | **t,tat =l * = #,, =, l== tncsilt - [
tdt = ron"r**f,[O-*)+d$-p) =t,tL_tz
-- tarcsint + JL - t,s2t-t.I = L-lz = - ocsint + itJ t - tz + i (r - t'a)i - torcsmt
-tr-*+c92Lt=!xatceos!dx.
TI
EaZl Hiss.-hissc intcqqllana dfMrunu efltiq edek:
u - *"*r'=, ar - +' :dt - au - -$,ae - xdr * u =Tx' J, -r4 '' r'tx'- t
Budr'r inicqaldt aazere elaq:uz- L Ll xdz xz 1 I r-=-----:
I = Z6ccos- - zl m= - arccos - -lix'- t + c.
e2J IYryd1(1-rr)tEdt Hisse-hisse fotoqrallama d0etrnrau te6iq edet:
I u= atccosx,f x(Eccosx I
I ;:6* =
lo, = -iu - ,)4a._atccosx t I arccosxa--,lt=F ! r-r' ,lT=7
du= -":,ll - xz
-xz)+ rt= +,ll - xt1 rr+ 1r-lzl-l + c.2 lx- Ll
I
926. I cif z dx.frUt htoqraldE fuokclymm dcnecesini aga$ salaq:
[ "r,ra, ='7[<t + chzx)z ik =il O . 2ch?x+chzx)dt =
:il 0 * r,ro, *ff)* =i(f,, *slzr + |sir+r)+ c.
e27. t = I.ltEar.E tfl. th, = t2 ovcz,lamesini ryaraq. fua alanq:
d -e-' - ez'-L - 1. 1+t2- t' -
-
= t' ) Z = - ln'-"""""=ct+e't - eu+l 2 l-t''l7-tz 4t - ztdtar=1.r*_rr.6:4dt=T-.l,
Ovulcoeai va nedceleri intoqrlda aeam alaq:t- t tzdt f t2 ilt
r = J,tau ax = z l r _ *= 2
J G_ ;i 6T;1=
= I L+ - *),, = l^l#l - *ctst + c,
t =i"l#f - -.tr(ftro) + ".
22t
Ineqrallot taprn:e2s.r:17#r.EaIIL x = 1*ezlemasini aparaq:
a, = -!at,I d, :t__i* ( t,J FG+fi = J F6;S=
- l ,, *,a' =
=-f ,u-r'z+r)dr .lh=-(f -:., +arctst=tl 1 1\ 1 '
= -krt - lr' + -) + arctg- + c'
ns. I#;FEaZL Kosrin msxrscini wruqlacr ayraq:1+ra+18 = (sa+ t)2 -xa = (xt -12 + 1)(xa+12 + 1).
lnteqrdaltl ftolsiyam sada kasrlere aynaq71
xa + x2 +L (xr -12 + lxx{ +12 + 1)1l xz+l l-x2 \
:-l-1-l2\r{+.r2+1'xa-x2+l)'
lnai tu itaami inte4altayaq:f dx lf r2+l lf l-x2J ;*r.# :
zJ ;t* r, *l,it + rJ r, _r, * rax =
.'**. * .!f j-+tu=x'+r+$ 2l x2-t+iz
=1f 'G-) -11 a('*1) :
't 1*-ly'*, " G*I)'-,1 r-_1 r lr*l-vel
= z,t,*"rs i _ _+," F;;tGl
* "
=:I
=_:,1+.i.J+.r.o1."
ttr.r=Jff.ffia,'fuIIt x -- sin, evazlamosioi 4dq' Oods 8ltnq: -
,=l*y #"=l;4fficostdt=-
= f fi t' .'"*'*'lo' = I ir,'r* * l,al- =
:1.lru"=t!r+u (1)- ;= I ro" );==L
*"==*,1=
: -rnO, * [nntdt = -tctgt t tnlsintl.
Bu axrrmcl naticoni (1) ala nazere alsaq, yazarq:
t2 t2I =l+It =Z-tctgt + lalsintl +c=
=-!@
eso. I-L,;777;g.fldll
r dz f ,2 d, I x"=t l_lffi=i;ffi=q46a:4{-
={#=l'=;-o'=-4*l=
1
= |(orcsinr)2 - @csit x' ctg(arcsinz, + tnlxl + c =
= llbrcsinx)z
-fi7 *r"t o+ tnlrl + c.
932. t = t x,l7T-ttntVz - ta1Ealll 12 + 1 ='t2 evezbmsisini ryoaq. fua alrq:
2zdt=Ztdt=xdz=t&.r2=t2- 1 ol&rErmdra x2-L=t2-2 olrc, Brmlan verilenitrgaldaoeaeahq:
r _ lft = J flx, + ttn!x2 - tdt = 1J t,tn(t, - z)dt =
ju=ln(tz-2),dv=t2dtl=l zt t3 l=
I au = ,z -re'' =; I
-i(f, u" -,> -i t *") :i u<* - zt -l t f *, * fi\* -=',
^u - zt -l(f, * 2,.#'" l-#D .. =
=l,nr,' - o -; -i' - ft ,"ffi* " =
-@P ^o,-'r - G;ir - i.tt., - ft"ffi#l..
e3l. I=f--L-- stu:Vl+cot-EN. l*cosx = tz
z = occos(tz - 1) = dx = ---4-._ , ,-=-o *t.= -#=Jl-(l'-U' 'lz'-t'2dC
.12-a'
5i2a = ,r,[-[sz, = h{Z -tr.Bunlm verilsn ireqaldr nezera ahq:
':[#:-l#--l#-,-'t#aSade kcsbra syrma
L A B Ct+D
-:-
tt2(t2 - 2) t' t2' t2 -2
evgzbmsini 4p6l(l.
istifrd. dclcA(ta - 2t) + B(tz - z) + ct' + Dtz
t2(t2 _ z)
(A+C =O,
t,_#:?-,f = o,B = -t,o =i,c = o,
f dt tdt t dtt =2 J tz6z-21= - Ja* I t'z 1=
L L lt-J?l 1 I 1l/I-+cosz-vz-- ,+ *tnl, * 121+
c = Jrr cosr - zJz'"l,l7:* "*i *,tzl*..
Qohpelar:
ett. tt-i),rffie
- .- 2
wz - tt'l2x-"13x) *
h*qnlbtfmflaytn:
tQ-xzf e.
!iG-ia*.I(t - rXl - 2rX1- 3r)d.
{!-\'*\r./
'(:.*.*Y.Ji-z{?+r.t_-{;______-4x.
t'-.,#'*.
c*ab:27x-ex3 .15 -if .
c"u"b, 625 *3 -125.x4 +30x5 -1916 *!*7 .337
ca'eb: .r - 3.r2 + 1l ,3 -3 ,4 .32
cavab: :-!-zkrlrl.
"o*, rAn-t-4x 2rz
cr"ra, ! *|li -4 *'{ rs *!# .5t73c**, -fr(r.|.jr.ir).
"**,\ff.cwaa:z*-lfiz*5 -1{r7.
232
931.
935.
936
,tr€u Cauru: rap-{.
Cavab'. x -arctgx.
o""r, -,.' jnlllca'ab: r.2hl+1.
Cavab: arcsinx + In(r* {* r').
c"*b, 4' *2. 6' * 9' .ln4 In6 ln9
2x+t _5r-r 2 (l\x I fl\,l' ' dx Cavab: --l'l+ l'1.' lox ln5 \5/ 5b2\2)
,'3' *l * carab: ]e2' - e' + x .'er+l z
j(l +sin r+coex)& Cavab: r-coer+sin.r.
tJt-timz* (o<x<z).
ca."uu' zJ7[1] * J7rg,,.{.*1-cos,}, (, = x- 1).l") | n )" 4'
lctgz xfu .
ltglr&.l@sll* + bchx)& .
lthzx&
1cth2 xdx
911 J
945 I
946 i
947. I
91E
949, !(28 +3x)2&
,)
:=_*1+ r'
*2
-.;dl-x'x2 +3 --;-dx'-l,h;?.^ll] ._--------:----.-
Jt-'a
951,
952
9t3.
951
955.
956
957,
95&
Carab: -r-clgr.
Cavab: -r+rgr.Cavab: acla + bslx .
Cavab: x-rlzx.
Cavab: x-ctl*.
a3
.,[t t-JPtI r--
! f{*+O*=!f1m+b)+c (a*o) pdittdan kqob enkagafuh inteyallan haabloyn :
le . Cavab: lnlr+aj.'x+o
cs*"b, -L (2, - 311 I
22'
961, l3Jvx*.&I-
9&.t56x-212
e63. ,!EF" c"*b, -;{fr:,T.&'2-3rz,&,12-3x'
959.
964
9&
gdff. I
969,
na
971.
!(2x-3)1otu
!(e-* + e-2*\&.
I (sir 5r - rin 5a)d ,
.&I*P.fl
.*'1+ccr.&'l-car
C-avab:
Csrab:
C-avab:
Cavab:
Cavab:
Csvab:
Carab:
4
-]o-re;'2
-------t5(5x -2)z
965.
967.
968,
f "l,"ti..,A;;l-7e-' +!r-2'1.
- rsiu 5a - I 06 5x.5
a"*t -!rtr(x*!\.2 -\ 4)
a*:al.a*,-wl.
9rz - !lsh(2r + l) + ch(2r - t)ft . ci!4rt: )pnp, +l) + sr(2x - l)l2
214
.&t--' -) Ich' -2.&t-_sh2 |,'
cavat: ztl.
ca*: -zc*1.
975.
9M,
,t:
9EZ
htqrutoU yaanA ptumilc aybatu wzqdtot Mloyot :
car"al: -il-x2.4
calab: i0+l)t.
.x&'^h-?
ftnJt*7a..th
3 -2x'.x&(l + .r')'.x&t 4**4'.&rA+r)G..t&lsm-'-;.xx'.&' *J7 -r'.&t !'
(x2 +t1i-rfut-t 3'
1rz -112.t*t2(}xt +2n3
c"'*, -fui:-zllovau: --L^
2(l+ x')
a*,l"nslCz'vab: 2arctgJi.
Carral: cm !.x
cavab: -arcshfr.
Cavab: -!.Jxz *t
C"r*, -*.'l'2 -t
c""b'L1,li7*n.t
971.
9n
979,
991.
992
993,
994.
995.
996.
gE7. 1xc'x2 tu.
gtl. r n'&
'2+er.tu
989, I , _,.
9s0. /n:'*.r
,&'xlnxlnflnr)
jsin'rcos rafo
. sil r'i*3,ItC*&lctgrdx.- sinr -
' Jcos 2x
. cos r' Jcos 2x
. slu' J ch2x
.&',i.2Ali€,'
toot. l'-;! .sin" r + 2cns' x
rmz iecos t
ru3. t+snx
Cavab:2arcsirrG
cavab'. -L"-" .n
Car,ab: ln(2 + er ) .
Cavab: arctge' .
cauab: th3 ,.3
Cavab: ln [n(n r)l .
Cavab: 1sin6 , .
6
Calab: -r-?- .
a/COS rCava!: - lnicos rj.Cav'ab: lDlsin ! .
l_cavab: - ;i hl.r2 cm r + ,.,G 2.i .
Cavab: {orcdnlJTsin ry.997.
998,
999,
Avat: frnl.ttrt*
+ J ct zii.
c^u^a, -!\Fa\.uwra: $anstffzlc"*b, h[s(;.;)
carab: hh;1.
236
.&986. I _- .-Jr(r- r)
.dxt-.' chx
. s/aorchtl: tbc .
",l sh4 x + ch4 x
_&'rhzrll&'f'"tq*7+x'
C-avab: 2arctgex .
1ao6
1N7.
lNS
tMt I<*"t^r12',[l-5.
illa t#* c*,,rt$(,.i)*=r(,-*)
crurb, L rotr*z -=1 ..12 * x42.
. *4&tul ,G{f,
. I l+xIUZ l---;ln:-'l-r" l-r
1oI3 l_=gl=_*Vz + cos zt
I0t1 h+iM'.-e Cavab: i*oro'*,sin'r+cos'x^r -.r
1013 lt't &.'gx - 4x
, l.r ^rlCavab: ' hl:-i-l2(n3 -ln2) l3r a 2rl
Aynlq oaduru a6fu ann*b ryfiffi iatqrqllan haabloyn:
1016 tx2(2-3x2)2& car*: JJ -',iS,1'rotz ,iI*
a"*, t=u( "h2J *,lino * * "ffi2.t2 \ 42
Cawb: t1[u r.
cawtt f,{arcw)z.
carau: --1-.arcsrn a
3
canu, Jrnztr*fi*21,
cevab: ---]-.t5(rr + 1)r
or"t']ro2|_,1I.
. cevab:Ar{,*E*l
CaYab: -x-2h[-rl.
237
Mt tLeruD. t#enza tff*1021. -2rfrma
cavat: ]tr- r)2 + ulr+rl.
Carrab: r + I4l + r21.
**,fitffi *znlp-?l-*.
**' {-{.*-*..-ti,.,l5432.l r 1l
O'*' ll(r+l)z -(r-1)z l.,L I
Gdoatt* , --le-s*>*!.1
t@!
t02z
IA6,
1027,
t029.
csvub: eed;Dt-a*".**".J
l* &'r+l
'J*t+JA!ilT--s*
ptlGT*,&'G-DG+3)
tfu' 62 +l)(x2 +2)'
&'1x2 -zy? +z'1
-rbr(r+2Xr+3)
o"*, -tiif'r2-sx11.74
Carob: fro * r2)5 - |o
* 1215.
c"r"r,Inltll.4lr+31
ervfi: anw-fr*"si
**#'l#-**"oic"*u' u l'*311.
(x+ 2)'
Bt
twa ry#J **,1^fi1031. tffi+p, 1a2 *b21'
*,"r, fi(i*asi-)*asi) tla * nr
1032 1sb2 rtu ovat: i-|srnzx.loii !*2 *b Caut: i*|u"2,.1034 Jsinrsin(r+a)d Camb: fco"o-|rin{2,+rl.
1035 Isin3x.sin5xd Caul: fsinZr-*1sintr,
1036 t*i.o;,i* c.'rt' t'n|-f';"f .
1037.
r',(z'-f)..*(r'.;)**'-**('..#).i*?.Y)
1038. Isia3 *a Cavat: -ccr+|oos3 r.
1039, J.*3r& Cawab: sinr-fsin3.r.
luo - !sm4 x& ovab, f r-]s" zx+
}llm4x.
IuI. !oosa x& o*.t, |r*ftit zr+ ];tintx.IUz 1ctg2* C-avab: -x -ctgr '
Iui. !ts3* car"ab: |rs2r+ntsrt.1014, lsmz 3x sitz 2x& .
cavab: -fr coo 2: - j.*+r* fr .* ex + fico.* + ficc,tx -
Iai pJL--. Gdllrr*t"l=sinzr+cos2r.sm- , cos- .r
Cawb'.tgx - ctgr .
.&1M6 J-.=;--
sln-roo8x.&IU7. J-:---- ram r cos- r,*3, ,lN& '.1o,
*
.&1N9. I 1oos'r
cavab: - .L + lnl,r(;. ;)cavEb. -J- + Inlrgxl
2cas" x
Cavab: rnlsin{- }sin2
x.
Iosa ,(t+e!12*Cavab: r +2orctger .
-I+!shzx.24?,fi,.3
t&1053. ' sh2x
"hzr
105L lshzx& Carrab:
1052 I slx sh2xh . Caub:
Cavab: rgx+frg3r
CavEb: - (fu + crhr).
Uy[un av*zJanobr qamuqla osagdatu intcqrallor hmblayn:4
1054. 1x2zS-x*. Carab: - 1rs - t2r+ l+x2)6 - r)f .140')^_
Carab: - _: (32 + Br +3xL).!2- x .
l)
caval: -f (g+t *2 +3ra\^ltj .
15'
cavab, -6i*T'? e-ssj
ca,"u, Jrs3,+]rs5'.
_2toss !f**{
Ilsd ! ^ =&tll - x'
tosz. !rs(z_sx\l a,
- sia2x -n'alaftcos - r
rose. tEg:.'r\il+lnx-dxl(Mq. l---:.Jl+e'
.arapJi &106r. J----F-'. .
{x l+x
-12il)62 t771dr.
')_Cavab: I (-2 + ln x)JI + hr x .
Cavab: : -2U1.[+ e'1
Glvat: brctgJ;f .
r-;--' .olduqla - Jxr -az +2atu1fif,i*J-r-o).G1statiS x + a = (b - o1sh2t.106&
J(x+dxx +D)
Cavab: r+a>0 vs x+D>0 olduqda 2h(fia+Jx+U);.r+a<0 rlo r+6<0 olduqda -251J-y-saJ=x-b1.
Ifis*hissa inteyollaru tsulanu aibiq anohla o$agdoltitueqrullor tqu:
1069, Il"rdr. Cavab: rflnr - l).
241
x = asin,t, x = atgt, x=asin2 t w buna o4o triqonndrik Cevintatardbiq drna*la ay|tdtht in eqnllafl lapn (pqamenlar masbadir):
avab:;,[t i.hl,.-ftrltml iliZ* c"*b,jrfu *j.oin,
catab, -3a r x Jiet=i *lo2
^r""- fiGMaiS x-o=(b-a)sin2r.
1M4 t*E-*' \2a'x106s.tffi
Cavab: 2 arcsin
1066 r[7 *t*)_
C.*"t, { Jr2 * 12 t 1-m1r* J o2 * ,21 .22t(N7. 1,f;:-'6lx+a r-; .
Cavab: x>a olduqde {:z -o' -Zoht(G-, +Jr+a1;t<-d
r07a ,lt:')'* gunrg' -111o2**2tnx+2).'l x ,r r1071. [^fitn2 xdr. Cavab, 2,1 6r,2,-1r",*fl.
1072. lxe-'ax. Cavab: - (r + l)?-;,
1073. ! xze-zxdx Cavab ' -46' ***11.
1074. lrte-,2ax, Cavfi, -'2 jl
"-*2 ,
1075. Jxcosx dt Cav*: xsinl+cosr'1076 lxsitxdx. Cavb tchx-shx.
1077. !x3ch3xdt cavs, [4. *l,rr' -(*, +)*trl.3 e) l.3 27)
107E Jarctgxdx. Cav*: xmctgx - !y411 t21 -
1079. ! tcsnxdx. Cav'5' ,*".io r*,',[*f .
10Ea ! x2 uuos xdx . a"".6 - -': " ,'17 * t *""o., .
nu. trydx. cavab -se1{Lx-r|+Oi
totz yt,,l**^fi**p. Caud, rlrr(r*r,[i7)-.il;;'.,66j. 1*t!-* Cavab,'-#ti-:
1ng4. Jsu.x-tn(rrW. CarO' r,]re jj "*, rrrr,.
inteqrallan taprn:
1085. 1xser3 dx. g"rr5, 11r'-1;r".
10Ed !x(arctg x)2 dx. Cu*b, {<*nwl2 - r*"tg, * L,1;.n *."21
10E7. !xzuryt+ dr ca"ab' -ll-ihl,-4-+"1[,ol
. rlr1, * /l *:2; ,l0E& t-----:-^-6.Jl+ r'
tole. nk--J*. crrrb, !J? -J *Luo" ff,
(o*o).
e.t*: i,l7i + 7"i,. .,i7i1.IB0. il? *" a,.
IogI. l12 r[o2 * ]at.
cuuuu,'(2'2,- "2 I i;4 -1a-^1* *,t7* ? 1.
10gz ixs;nzxdx. a**, " -;r-rr--t"
Io!t3. I"&a Gtvaa: zdi-tyE.1094. !xsin{xdx. Cavab: z(o - r)G*.rir -oiz - ryrt Ji.
eqfclgx ^ . tl+ rteq@1095. i'--jd c'u"U' t;fu
(1+ 12.1i
tmd icos(hr)dr Cavat: j tsrntln :) + cos(ln r)l .
10s7. le*e,,bx& &,'an'!9\!!4r*
lw& 1e2* suz x & c^ar"u, f,tz - ri, 2x - cos2x).
1099. g+"--* cavab: - r + f hr( | + e2x1- e-xarctglerl .
. In(sin x t -
l tklt ,;;;'* Cavab: -.{r+asx.tn(esin})}.
, xdt Cavab: .rrgr +lnccxl.1101. J 1-
cos- x
-.--;Cavab: Vl+ x' ln(r +.Vl +.r'; - x.
243
Ilovbdi inteqrullotn haablonmanr hva&d aghadfrnin kanoaih phlagatblmrdrra v, agdgldakt da$u ara acaslant:
_, & I rI l=== = :arctg' +C (a *0).'a'+x' a a
, t*=)nlt:_'|c a*ot
ru.1*.=rlnla2tx2l+ct a'+t' 2 I
Y. t-4===arcsinl+C (a > o).t Joz -x2 a
w. l#=tJVx*+c (a>o)
vn. !^tnlax =;,[E? .t-arcsinl+c @ >o)
1106, i3sin2 r - 8 sin rcos x + 5 coo
2 r '
2U
wa !.[?T?* =!J?;7 r+r,f,* J,' tdl*c (a >o).
IIo2 ! &- lob*o).
a+bx'
C2vlab: ab>o ^u** fi**ref,'ttaD<o or<rrrq<ra ffirffi#]
-&11011=;--.' 3x' -2x -l,(x+l)&
1r04 ,7+;
, ,3&Il0t t-T i -x -x- + I
c"rr"u' lnl '-l l.4l3r+llcavab: lh(x2 * t *l\* L or"tt
b *l2 J3 'T-
cavab, lL,tx4 - 12 +2\+ ):o 2x2 -l4 2J7'"tg-i-'
l. l3sin.r - 5cosx- lnl . --2 I sinr-cosx
tIoT' Y-9-- @*o)^!a +bxt
110& I
II09,
tlIA
1T1L
1112
1114. I
II1I. I
&1l-2x - f
.&,,12x'-x+2
.*'.,1s;r-r,-xfu' J;3t -2x4, ,3&'J;4-;x\
Cavab: 6 >0 olduqaa,[m(r.!a *'[o *t]1;
a>O Eo b<O ,ld,rqd, #rrc""C[-5.Cavab: arcsinf .
I-----' 1 I 2x-lLalvED: -V)+r-r- + -arcsrq:2 Jzt
I 4r2 +3(A b:
-arcsrn-.2.12 ^1t7
co,ro, )Jl -zt t.lily'1-t."F -z,?_rl
-nl,*l-z.F*Jl.&1113. !---:'
x{x'+x+l&
G"F
1116 j
1117. I
Cavab:
Cavab: h4.Al,., I
{r*z)zG *zr-s
**, iE#=*}."..ffi (p+{>Gy
^ 2x-l t- ) 9 .2x-lLa b:
-\i 2+4t3
ut
z+x-x2dx
x4 + zr2 -l rdx.
&
,[i;{i-;r,,1., **,[t *z] -tl.
TlIE
c"""u,- Jr * J. ],*i,f -ni="+El,F il.f ,
Qcyri-n&ayyan amtdllo asulun t ,2rliq atmhb aSa$rdokr
teqrallon tqn:
rllg JilJl.p Cavab:lnlr-2|+lulr+51.
rr2a t:'oq-.x'+x-2^ , re ,a 3r7 5*6 llr5 - 2lr4 * 4313 -g1l]r171r. trl '-] =,1LawD: --T*7"6* s- 4- ' 2 -,,* -t-lfij-62Ai
rl2r. !,"*r' &x" - 5x" +6x
ovat:, + jrd! -9r"i, -zl* ?!r"i"- I.
cavab: - 1 .?r"E-11.,(r - l) 9 l.r+21
6rr"r, J_* jrol,2 -rl
fi24 Iir + t)(r + :;2 (r + 3)3
xh1122 l-;-r'-3x+2
x2 +LIl23 l----d.(r+l)'(x-l)
1125 I
1126 ll*t'^*o *x'+5x'+4
Cavab: ,r"tgr rs,ln!:!- 6 rz+4
24
Car,ab: 9r'? {orlil* t,kr + txr+ z)161.
46+2)(x+3)2 t | {r+3;17 |
*5 **4 - zr3 -2x2 +x+tgur^6. _ 3x2 +3x -2 - * 3 rr.,lr*ll.
8(r-lxr+l)2 l6 ix-ll
l-x+x2_-6'rvl+t-r'
1127. I
TT?E. I
1129. J
1a2 -lx + +11x2 - 4r+ 5)I
Cauab: -- -; - arctg(x - 2') .
x&(x -l)2 (xz + 2x +2)
cavab: - - -L-*]r" 9-ti' - ^!arrg(r,l;.5(r - l) 5U x'+2x+2 25
,(l+rll1+r+t')carau' rof-i-,i - a or.tpt
+ 2x .
il+rl J3 - J3
rt3o. t# o'"r'|r.fS-i'*of'tlt. th ** i"l!l- i.*no
_&1132 l-r---;-.
x* +r'+[.b
il33. ,a;"*7r.+o'o' ]r,jffl.*,,*o#
Cayab: - | - +lh (l+r)2- +ldE *- t=oon2':l .
6(l+r) 6 t-x+t2 2 - 3.i3 {3
OilToydshi Asfunu dbiq aaohla aga$daht inleqrollot tapa:
t ,],4. J -#---a. c ' tt 1r2 + rt l(r- +l), avab:
G +l; + EEctS'x
'
, ,2& Car"ab: ---I- +arcts(t+l\.1137 t G\;;* x2 +2x+Z
. x2 +3r -21136 l.--_-*(x-l[r'+x+l)'calat: --!t+l-*16 (x-l)z * 8 o,.crp2x:| .
3112 *x+l) 9 12+r+t 3J3 " '/3, & -, 7r5 -lL. 2t .lx-tl 2t
1137. I G4_D3 cavat:ffi + tz'htlx +t - Aalctsx
&
Atagrfuk, iateqrallsnn cabi hissadz,i aytn r:
,38. 1 !:l_-a,(r'+rz +l)"
ca'tab: 6(fr3+D
,dt1139. J ,- ---.(r"+x+l)'
Ex4+8r2+4r-lLavaD:--.28(xr+x+l)z
Cavab: - -=-I- .
x) +r+l. 4r5-l
1140. I -1---1 "tfr.(.r- + x+ t)-
MLtufif asau@ tarbilt drrohld agagdahr intqrauot tapm:
_3rr4r. Ia:;-o"t33ruawo: - *,-fr -
ez1,:rro - r*;5or -
rrG -,,1142. f? * **.' ro+l
cr*b, Iu (I'*=l)' *!***3 * | *.*2":l .t2 xa_r.+l 3 " 2,8 " Jr
rr43. t:ne. c.*u,-1[,]s * r rl']-t/ib'1rr0-19;2 loolrlo-lo zJro-lr5*Jro. ,9&
rr44. Jaio;r;r.- xln-l ds
rr4s. t6a17
,&I 146 ,
4rlo * 2y
.&l_I147. ' x1:10 ..112
,4 -l,(r4-5Xr5-5r+l)
ca\rab: -rooi;*., -Larcrl@s +t).
a,ot, |(*,u." -j;) ,*0,.
c"rat, fir"jo ^
cav*:ffi-rtt#curo, 161 'J'a - sl
I5lr)-Sx+tl1148. I &.
24t
rr49. t
1150.
1I5L
1153.
1t54.
x2 -l,4 *13 +? tr+l
lue4rulah funkiyalan rusiuol funhsiloya g?rinnh ay&MtinEqrull@rlqu:.&'t*G'
I 2x2 + A' .fi)x + 2LarAD: --E rn - .;-- -----:-- .
V) 2x' + (l + ./5 ).r + 2
Cavab: 2J; -2h0+ 'fr).
**b'i'Facffi;,ltrI 3 4{6-l--ttctg-,zJ't ' ,l't
.l-..8+l .lI52 l------;'===ft'l+{.r+l
c.avlrbi 6t -3t2 -2P *|ta *lf -ll +tuo+rz)-6scttt, t=Q,lii..d,t{rr111r'1-' c"*,#_#,.Jr+t-Jr-l .t-or'Jr+l+Jx-l **, + -
q. ir,,l,.-..[t- rl.
uss ry{- P.01lir- (a - r)
cavab:- I,.fr":##.,#rts6 t--3-: o*n* r=(u'-'\'.'l+J:+Jl+x I 2" )
ovab: i + J; -|fir ;l- lr"(6. 6;1
,s7. l-+: ."o0, -rlr-"*rfill.(l+r)Vr"+r+l I r+r I
rls& I 4------ carab: '*,t3.lin{-'J(r+lhil-r-l ,./5 J l+x
249
x1t+ 2Ji +3Ji)
lrse. 1'={*{I+.r-r"
t16A l "oL- Jt*12
(r-D3[2+3r+l'
.tutr64 t;;DtJfr
665; JIf .p* 2" - i,,"r. # barada ki .r < -2 ve ya x > 0.
ffi ,An ot furd+oonrn sob ha*tara uyvtrtil" I #!eintc$ohru toptn, burado y = "la2 +ix + , ,
I16l. I(x2-ty.[i-i',-:
- t,.ilr*t-2,,f7-r1I r-3 l.l3x+l-2,,1x'-x--arcs[.........---lnt2 lx-llv's 2l r+l
cu*b' 3'-s =''Flr'*t - "="ieru.,r:ar!'.*-'i2o{x - t)t 4oJ5 | x-l
I
Carab:2
**0, [#,- #,3 * ft ,5 -#'. *JJ;7 - fr r,(,,./,r,,')
-t3 -6*2 +lLr-6 -
I16I. l__77::-e,J xt + 4x +3
**0, [*-
-?. ")A; r,., -*r"[. 2.,,F.r,. :1.
,--! -1162 ' ,oJj _t' l,.'ua,'!4,8-,
1163. I
tr66 ,F-':'*(x + l)'
1167. I
lbe I
tbs. t
i*1*.1.[1i1'c"*r, - E J * z, J - zarcsinll - lrr.".. tJ?
2 ,12 ^12 [*rlx&
(r2 -z* r z)^k - a,t * 3
"r*, -EIL-r"**# (r<l voyr x>3).
(t+ x2),lv *2
t rJllAvab'. -- alct"
-
.
^12 'zrlt. x2
.&1170. !- " '-r=:
(r'+l)Vx'-l
1171. JG*tfKvabo, aChadlhi kononik pkla gxirrmkla atalhdth inrqrol,
hesdloya:
(r + l)rbtr72 . ri-(r'+r+l){x" +x+l
. (avab: _*:!_3Jr2+.r+t
D ^[;t +b,+c =tJix + z,
4 ,[F *tx*" =o*J;,3) b(r - rr{r - rr) = 211- 1,;
Eyler avatbmalarinl lztbiq abhb aga$dak inleqrollan tapn:
rtzs. 1,,[7-iJ*
Cavab:
carab : ---i- + + t l'[7 :ro I
zit*xz l^lz l,r!*12 _rrtl
a>0 olduqda;
c >O olduqda;
251
**, - q:i' * b(,*1 *.f,t;). i'ffiI
I
c*"t, |{}rr, -,13 + 1z - )-3 1 + 112 - r12 - 1z - 11-21 +
1175. I
+ [(z - r) + (z - r)-r]]. itlz - ,,
1174 r;ft*haradaki, ,=r*J;'-2;.
J,, -r',*Irrl'*l'-tf|. 4 lr*Jr2 _rlI
Cavab: J + x - Jl- r - liarcsinx.
t*x*z[*r* ](z***z,ll* r* ?)2
gun5. 2(3-42) . lLl]4t?al, haradakr z = -x+ Jr(l+,.5<t-, - "2)
5J5 lJ5-l-2zl
Matalif awllor otbiq drroLla a;agdaht intqmllan topn:
dr
m_o=.,car"ot, ftf2*r*
tt76 V#A:*.rr77 F1w,
r-..----;lCavab: Jl+x+x'+:ln
1r7& 1 Fn:*-../r+Jr+l
*"*. 1" t2(z'2 *t* 2J'a *'2 * I2 x2 +2+2Jx4 +x2 +l
Cavab: r > 0 oldrqda j1(;;f tt1f,i7 * ]
r"rJi * J-r * O
c"*r, l[r, - ui .,i]- 3[,,.,,; -,;1
, (?+Ddx o"ro,- ,=rlrf .=ffi|.117e. ta_;Fi *"-.-E-l--7,
I
IIEO.
1181.
x5&lIEZ l=::-=\it - r'.&
1tE3. t177!l+J-
.dtrE4. ,m7
c,"" +"]+l-!o,",g,,4 lz-l' Z
.&ilts. ,m
c.avas, -z+!) -1, **rr='ll-? -
harddaki ,={;V.x
=187x**, l^l$ - i,**'#, twat'k . z
t &- cavrh:2ra -!r'. uoa"u. ,=drJ.ttE6'rr:/r.1 4 e Y x
\t x
Jsin^ x cos' xdx gahlinde otan intqrallo eynilik pvintolarinin
*omayi va ya drxani azaltns. fuw annn tdbiti ill hesablarut,
burutb n v) ruun ededlardr' Inteqrallor lqtn:
Cavab: sin x - 3 si.o3, * !rir5 r.35ca,rrr, lx - lsin zr+ 1ri, +r+ l.io3 z,--'- 16 4-- u 48
Carol, lr+f .io zr + lsin+x-lsin3zr-'* 16-' 4--- u- 4E
- r sin 4r sin3 2x+--::-.t6 64 4E
.*,s2x 6J 2x 9o85 2x(arraD: --a *- % ' no
carar:-Jcosx- #'u+r;l
o,au, }rr1- j u'*!' ft*"ot|.u.a"n,,={u,'.
1187. lcnss xd.
1[E& 1s;n6 xfu,
1189. Icc6 r&
1190. 1s;n2 xccx,a xh
1DL Isin5:cos5:dr
.*4,1192 I.t d
s[t .r
.dr119J. J -1-
c(x- r.dx94. I -_;- .sin' r cos' x,&
1195 J. - asll] IcGl x
lIgA 1ctg6x&.4
ils7. fg#acos - x.dx
119& J'-l:=cos ri/sin "x
ca,ab: th f * lnlrr(l. r)l2coszr 2 l"\2 4)
o*u, |'[ffi]+-".#, haradak, r =3Gi,,,.
.dx1199. J;:.
".ltgt
a*a, ln*!\tt.***#, harada kr, , =W .
Navbdi intqtolh,
t. sin a sn g = )tcns(a - p) - cos(a + p)1,
II. cos acos B = ![cos(a - /) + cos(a + /)],
III. sin a oos f = )[sin(a - l) + sin(a + p)]
{,nua6,'s!.a * "t' - cry:J * jk.y*l".
422c.*b,*.*;."|n;lc^u^b, _
" _"rgrt * *4!,_"ro
*u*,'l'
- f *.4r - 1"o. rr.Et2.r sin 2r sin 4,r sin 6r4 8 16 24
dtsturlonau kanoyi ilo haabhnr, Ineqrullan tqn:
1200. jsin 5rcos r& Cavab;
I20L jcos rcos 2xcos 3rdr Carab:
1202. Jsin rsin(x + a)sin(x + b)dx .
Cavab: -1co(a -6;cosr- 1"os1r + o+ 6) + l oos(3r+ a + D).
1203, 1crlr2 * a"2 brfu.^ . r sin2c sin 26r sb2(a-D)r s,.D2(a+6)rLa\/aD _+--f f_--:----.--'-- 4 Ea 86 l6(o-b) l6(a+b)
Nubati iateqrallarL sin(a - P) = sinl(x + a) - (x + B)1,II. cos(a - B) = cast(x+ a) -(r+ f)l
eyilihlarini tatbiq amahta haoblanv. integallan hesahloyn:
1204. 1.. + -
6uo6, -l-,-1,,i"f i$l (cos(a-a)+o).' sin(r + 4)cos(r + r) r':o{a - 6) lco(r+D)l
t2oi J*<r;rfo,-r, c"'"t' *1fu hlH[3 (sin(a-D)*0).
1206 I d*@sr+cos4
1207. Jtgxtg(t + a)tu
'dx120& J=--j.. Cavab:(z+cosrJsmr
- sin 2 x1209. I-:--4.srnr+zcosr
JRlsin x,cos x)dx v*liada olan incqmllu amani haHl
ts ! = t naztamasinin hamayi ila rusional lunksiyolann"2hteqrallanmasn a gatii lh, buruda R-rar,iotwl fu nlsiyafi n
a) Og* X(-sin r,cos r) =-R(snr,cosx) v? yo
X(sin r,- cos x) = -R(sin r,cos x) olada, ondo cos .r = r v, JauySun oloraq srmx = t a'azlamasini albiq ama* aberislidir.
b) OSar R(- sn x,- cos r) = r{ (sin r, c e r) olosa, ada tgr = tavazlenashi tatbiq andh lbe?iglidit inteqrullat topu:
I x- al. lcos
-6xyx6, L6l-2-l (sin a * 0).sina I x+ ol
l"*21lt
Cavab: -x+ctza hl "1' ,1 (sina*o).- lcos(r + a)i
I, (l*cosxX2+cosx)2
6 (l + cos r)r
cavab: - |t2sin, + *',. ;h +(; - *r")
1216 I
t21t I
I2I2 I
I2tt
12r1
q2 dor? ,*b2 "*2
,cros2 xdx
sin2 r__.,_al+sin'x
&
' (asin r + Dcos r12 '
- sinr&l-' ,in3, + cos3 r'
(o2 sm2 x + bz c.a2 x)2
&
obZ\-t z I @CavaD; -* + --'----._ arcIE
=-(aazL +b') 2ab" b(ab + 0), lwadak:., z = tE, .
@sra(asinr+6cosr)
-tk1217 l- ,
^srn'.r+oos'r- .ir2 r cos2 r -
1216 J-.a r d'srn- r+cos .t
6r.,15 -]t("io""c')2 - ]-*"r*(2*Y -"h'\6 l-sitrrosx ./3 -\ .J3sinr l
a""b, +r*",c(ff)
cavab: - t? * J-or"1rtfi4(tg2x+2) 4Jr''-'" Ji
buruh A, B, C+tbitbdir;a,sin x + b, cos.r = /(a sin x + D cos x) + 8(a cos r - & sin r).
inteqrallot tapn:
12tL J- "i"a ,k91tr,E-JCOSr
f '' :l' il?l & = Ax + Btnp sir.x +b1msr1| + c,, osrnr+Dc(xir
Cavab: 0,Lr + 0,3lnlsm r - 3 c,os xi.
tdrl2II ' qr^2 r+2*"2 r12
Ntubati i*qrallot
Grvan: , - lrarrtgGf2rs*1
.
curou, lontg(otF)aD \0 )
Cavab: -
a"*, !{$. Tz * ",t ffi - .,t - n * ". #r}
r2ts. t# o"*' fi-|r'Esinr+3cosri'
far shr+6r ocr+cl & = Ax+ Bhhsimx+bcax +cltcl &
r 4sinr+tc6.r+c ' ' Jasinx+Dcosx+c
(burda A, B, C-naalyan sbil anrallobr) olnwndan itilada edankogagdah intcqrullan tqu :
- sin rl22o lE*"tor*o*r^'
*"*, ;- ;,t(;- f) - ir(,.,rn r + cosx)
- 2sinr + co6r1221. ,l*r*;;;-
Ogat P(r)-n b-ao$ prtadMirs, onda
1 r<,v* * = "*1ry -'# - .eD' 4P] - c.
Dlwrdot funilab cbrah aSagfuh iatqrallot tqn:
t22z 1r3e3,tu. *"*,,r,[+-+.+ +)1221 loz -2x+ZV-'& CawA: -e''lx2 +Zl
Ogat P(r)-n baoi prtedlitfiw, ondt
! r {i *o" u* =s\Gl, o, - r?. n# - )..Tlror-':a.#- ).,
9A
! r61sa max = -Tlr ",'
fS .'+ - l.-'rgf"t,t -'P *t)g - f , c.
a' L o' tt' IDasturlanndan istilada edan* aSa(tdah inteqrulldn ldgn:
1224. Jrs sin 5r&.
**, -[4-4.?n'] - ('a D'2 :+ I
[) t) ";J*"*1T * +;Tistnst
1225. !(l+ x2)2 ens x & .
Cavab: (21.lox2 +r41,inr - 12Or--4r3;oosr.
t2z6 1x2eJ;&.Cavab: zetls - 5ta +2ot3 -60t2 +120t -no), harada ki, l=G.
1227. 1e**s?bxdx. c"*u ,*f -l -'*^zg-4.t:b.1-'-" - lZ" 21a2 +4b21 lI22E 1e* sn3 bx&.
- e^ f \( atimbx - boosbrl asin3D.r-3Doos3bxlcavaD.
-i---;- ^ -------
. -_-;- i.4 L a' + b' a'+9b" )
1229. 1x2e' cc x &.
cr"*:'j tr2 trio r + cos x) - 2x sin r + (sin x - cos r)1.
tBA !xe'sn? xdt.
Carab "'[t] - 1011(2sin2r+c,s
2:r1+11(4sin 2x *3cos 2x)].
1231. J(x - sin r)3 d*.
cavab: 1,a, 1,2
* 2,2 *"' - \.,-, * |,*r,) - (s*r, * i*, z,) - 1oor,.
Ata*Tdah, inleqfillat toP.n:
uz l(:fu' cawb:x+;\-ln(1+er)
258
dx1233. t?\r -:r'
I.l+e2l--'--;e.1234. I l]'ll+€4 I(.)-dr
1235. l____.r/e' - 1
Cavab: - I * lrnb, -rl, ]rnra, +2.i.21i 16
oCavab x +
-'-.I1+ e4
Carab: - r".",,,[,-;]
1236. , 'lTlqr' a d..
G"aU'.,1"h*+t-r * zhlr' * z * rF\ +",_ti - rr"* &1 Ie' .15
1237. ! o' , dr.(r+ D'
123& 1r3 ln3 xa.
t2rs. rl,!rl'a.'\ x /1240. lxarag(x +l)dx.
1241. ! ^fxarctg,ti dx.
1242 farcsin .lf dr
1243. j*cshlE*.- atccos.r ,I -- - -- --=-dx
1244. ' t0- 1211
cauab, "x-rl
6"n"6, 11 lur r - 1rrz, * lrn, - iz)Carab: - {G3 r* lu' r* l-u, * il.ll
Carrat: - j + jrrA 2 + 2r + 21 + t arctslx + t1.
carau, -r* ]u{ | + i *2J!' rrrsr.1; .
cavab' L,E - rz * [, - ] )u,.r;o ,',a I )ia\
Carab: 2ll - ./.r'l* (l * ,)arcsio 2"J
.I I ' ' l+x
6"ur5. r{cctr
- rn f- 12 .
Jt-12
259
ln f(x), arctgf(x), arcsin /(r), arc.cos /(r) lunhsiyalm tuxilohn qa[rdah inteyallan tapw, burada f (x)- cabi frnhsiyodr:
tXl I xarctgxh(l+ x2)&.
cauab. t - ocsr +(* ** - ;)** l) - rl .
1216 lrr,'l-&. 6u*.,-l:1161*r.2 l-x,t*,8)*.
1247.cav.al: -ufi7*ft r"t' - { -'1.
Q + ,211
IlwMillunhsralo doril olan truqnllon tqt:
I 24t 1 sh2 vh2 x& .
IUg. 1sh3xh.
l25A lshx sh2x sh3xh.
I2SI. ltht&.
I2S2 !ch2x*.
t253. t &'sla+2clt
r2s5 tru-!*
1256 '
cM'3sb-4cfu
1257. I shm snbx & .
l2st I shax cosbx & .
x sh4xLavalr: --+--.832
.1^ ch'xUavab
- - crr.
J
^ ch6x ch4x ch2x
24 16ECavab: ln cftx .
Cavab: x-cthx.
I
Caw.b: ftarctss-i
zrr l + t).2)12s1. #;#;;E a,var, rt*astu';2
cavab:4:arctrlt5)3"/l l "[ ./r I
.J
orat, - |r - f-u l3shx - achxl.
^ a chc sinbx-b shuccsbxLal'aD:-^-------il-_.a'+b'
Cavab:
2A
a chm cs bt + b shc sin bx
a2 +b2
MItufrf Asullan ffi\ ootoHz eSafuib laleqth tapn:
r2ss.t#f "*",i$-*.{iij
25t
IV FASTLu0owex TNTEQRAL
Sr. MOayyaN bffEAML caMIN LbrIrI rtMIt. Rhnan huqrub Farz edek ki, /(x) firntsiyasr [a,t] prCssmda
reyin olunmugdur. Bu parfet her hansr qayds ilefl =,Yo < xL< "'< r, = b kimi n hisssaya b6lah
Ar1 = x1a1 --Yi,) = nnxl,xi
kimi igare edak. Her bir [.r1,x1*, ]( i = 0,r,.. ,n - r) pargasr daxilinda
{i e [n , b] g0turak va aFBrdalo kimi _cam
dtlzaldek:
s, = f xe,ra,,
Bu cem inteqral cemi adlann. B0lltnnadan aslh otnadm i = lim;.-e S,,
ccminin sonlu limfi varsr, fua [$) funksiyasmm [n,b] parqasnrda
mooyyan inteqrah deyilir vs 1b
1 = | 114ax
kimi igue otunur.Bu terifder aydm olur kt, egar f(x) funksiyasr [a,b] pargasmda
maMud deyilse, onun bu pargada Rimm metada inteqmh yoxdur.2, Aga*tw yntun bteqnl Dafiu canbrl
f;/ firntsiyasrmn [a, l] pargasmda asalr ve ]uxan inteqral Dsxbu cemlerin-l n-l\- - ra'
5"= LmiA'(' 't" = LM'tx"i-0 i=0
geklinda olur. Harada ki,n, = . inf .{/(x).}. M, = sup {/(x)}\€lr.-\.-1. .re[.r,.t *r.
3. intcyallam kr *iyasl.1.r/ funksiyasrnm [o, bl pagasmdE iateqnllanan oknasr 0Sr zeruri vskd o"rt n-r
|l1ts, - t) = )'SI ar,a'r' = o
tertinin 0denmesidir, harada ki, <,.r; = M; - m. f(a) fimlsiyasmrn [o, b]porgasmda reqsidir.
1260. [-1,4] pargslllnr n boraber hisseye b0llh va
((i = 0,1,..., n - 1) arqumnainin qiymetbi bu pargalom ortslarmr
got{hmekle f (x) = t + r funksiyasmrn s,, inteqral cemini trym.
ZaZi Bu parga 5 vahiddir. Ona gore A.x1 = ! qarta gOre
f, = l;* (i = 0,1,....n - 1), x, = -t - t"it'.
f({,)= t+{,.Otrds
," = ;r;(, . l) = f [{;! - ]l =,,,,x=0
1261,
['r'a,inteqratmr inteqral oeminin limiti kimi hesablapa.
Hcltl l-7;z|Wry*mda fi-ni prgom sol terefindan gOflbek Onda
' =;I(-,. #l =:I(,-#.5)' = i[,,- ":; "]S=limS,,=3.
s2. QEynl-MoAWAN btrEQMrJN xoMOrt ILA M0AWANNTEOnAr.Ir,t flF&18 LttNItIASI
l.Nynbrt-L.lbnb dlNrou.Oger F(r) funkiyasr [a,b] pergasmda kesilmcz 1r(x) firnksiyasrmn ibtidai
funksiyasrdrsa, onda onun qiymati
.lo, f {r)a, = F(lr) - r(a) : r(r)ll (l)kimi hesablanr.
2 Ekre-h&sa haqzllanndMwuOg* /(r) va gQ) e [a, b] i.sa mda- " tt rb
I f?)s'G)a* = fk)s(x)ll- | g(x)f'(.x)dxJ"' lo
kimi hesablmr.3. Dqlpnla artzoluenos
Tu@ ki, aga$dakr gertler 0dmir:1. /(x) funksiyasr [a. b] porcasnda k silmezdir ;
2.d<t= p porgasmda rayin olusmut .r:9(t) fiurksiyasmrnpargoda kcBilmez tommesi varss ;
3. s(a) = a,s(B) = b.
Bu prtlor daxilindarb r B
I f!)dx=l ft-sk)ts'todt.td Jd
d0sturu do[nrdur.I2d2. Nyuton-l,eybnis dGturundan istifade ederak hesablayn:
t
fi dx
J-r ''62:fr.Ai
IfldxIftrn
J-a tf:;, = nlcsinxl'+ = arcsin- * arcsinr= ? '- = -'lZf:. ffesattay-,
fi tlx
J" ,r'sir''r' + b'c^',Ecll* arctg + arctgL = ] aO"u--a- istifade ed.k Aydmdn ki,
ilI[z dx t [, i;PiI
" ;;;F * + b+-o'\ = * J " W+;*
t ti d(tsx)=".t" d,_ef
=
= j.i.,*ffi f
a, = \ ar*s(+)i.., =
= n],f"'nn (i "i) - *un (i' ,'q = * ; = #
1261. l:lesablrym:
l' d*J -, x2 - 2xcosa * I (0<a<zr).
2&
Ea[l Funksiya llzerinda gevirmaler ryrq:[t d(x - cosd)
= l, = _ ,.' - i:"jJ - .oro rl =J-r(x -rosa)2 - sin2a I *=i, t= 1-cosc I
I L-cos.t dt 1 t 1t
-"""-:-----;--- - - Arctg-l =
J-.r+cosult' + sln'c( slll(t sln(l_,1*.or.r,
1 1- cosn 1 -(1 + cos()= --:-- arctg --.- - ---:- at'clq
-
=slna - sln(z sllld sltl(
1 l-casn 1
= -
ot ctg -------:- -t ---- arcta
-
=sina - slnd srnat srllo
I r( tr flr Ir=_t_r___!sina\2 2 ?, 2sina
Buradrr-.osn :rin2{ -. o. nnrctg-- = nrctg;;T;a = arctg(t9 r)=:,
r+cosa ,.or,{ -.- !- n aorctg
- = arct I ;;1j4 = arctgctq ;)= 2 - 1
olduEu nozore 8lfim4dr.
M ayyan inteqrahn knnayi ib asafuh camlarin limitini tqut:r265. 7 2 n-l
lim ( -1 + -= + "' + ---- '-)'n-e ,l' ,l- ,l-Helll
1+ it"=;L;melumdur. S, cami [0,1] pargssmda lr(r) = x fimksiyasmm camidir.
Odur ki,
1 + cosn
iL r'l' 1
itt,.l t, = Jo,or = Tl,= ,
r26.L71c =_n+l n+?' 'lt+n'
rrcIIL lH l,, =;L , ,_
i=r r - ttmalumdur. Sn cami [0, r] pargasrnda jr(x) = f t ot"iy*rnn cemidir.
Odur ki,tr dx
!,:Xt^ = [ *. = lnlr -.rlli = Irrz.
1267. tnnnlinr [ -':-----""= -.' --li-i\n:+ 12 rtz +Z2 n2 +n2 I
Hcllir3 ls^=-)
-_:-,"rl r * (i)-oldutundnn yazanq:
ft 7 n
iY,ls' = l" r - ,. d' -- arctgxlt -' ;'
126&Ll n 2n (n- 1)r\
lirn -l sin- + sin- + .,.+ sin l,-*n\ n n n IEelll
n _r
5" = 1|,i,1u=, ,r
oldutundan yazanq:
t'll'1 zlirn S, = | sinrxdx = --cosnxl = -:(-f - 1) =:.a-, J9 ,t to T tr
1269.
26
,,,1/ j=* t;. ... ,i-\'t' n\..l n 1 n \ 'lEclll
,,=}I]Goldugmdan yazanq:
jEs,, = !o'
u*ar:1,, -rtl,:;(rt-,) =JG'z- tt
Y ksak t'arfiMan sowuz kigilanlari ataraq, alaNah camlarin limitinitcpn:
1270.
I.* [(, . 11',, 1 * ( r * i),i, 1 *'' ( r + 11),,,, q-*,!t]
Helllkn kn /k3\sin'j=i. rh/ (k = 1,2...,n - 1)
oldulundan yazanq:
i (, .:)',,5 = i (, .:),,,#
jlrs, = pl lIi(' -l)= , [',,, -:)ri.r =o-- n ?- rtt
= '(i. +)1. = "(:-i) = TI27I. Hisse-hissa inteqrallama dosturunu tatbiq edersk hesablaym:
t')o
artcosxdx '
Hclll. lu = arccosx,dt' = dx I
[o' orrrorrd,=
lr_ = -#," = rl =
lr x lrr 2x= xarc.osxli * l, r.1=;,
dt = arccosl * i), ,---=d* =
l ird(l-.Y2).-ti" ,r -.= * =- il"-':;-i411-'z;=
1272. Dayiseni avez ehekl, inteqrall tapm:3fi dx
Js 1,r - 1;t?t+ r'Eelli
LLlx-l=--x=--l=dx=-.=dt,ttx=0, t= 1,
34r=i.t=i.
1
=-(0*r)=10
ztz -2t +lrl 12 1 2r:+l=l__ll -l= ___+Z=\t / tz tBunlm verilsn inteqralda nezara alaq:
-1 -!,r dI 11I --:
IJo (x + r)rI?lJ .,r
f:
-io,-i--:i\ztz - 2t - to (,-l)
r 'i---r." rFri(t-zi *+
1
6
t
,lr=-=t"lr-r+
,lor= -=r"ll-l+
2A
,E-i:1.+,1, ;-,t-*;l=
7= "=ltt
r:2
1= --= ln
r21
= '--= lnt2l#*l
7U + \A\zl0+vZ t=
r s+lvZ= -ln-.s27
1273.ft xdx!= |
-
J-rr:-x+l'EA
fr xdx lfLZx+l-l| _=_l _--cr =!-1x2 - x + I 2J-r.r: +x + l--
il,=:t_,d(x2+x+l) ,G.+)
r t' r x+|;r:1t"1*' * r* 1ll_, -*o"r.rrfl_, =
zI I 3 1 l-1r
=;ln3 - =arctg;
+;arctg (--=J =
1 ltt Ltt I r 1 In3 1r
2--- 13 3 r3 5 Z 2 13 Z 2t31271.
,= I'"Ea[f teteqralaltr frnksiyalr
x2+x+1
sint x + cosl x
i feriodlu o[nu8undm
sin{x + cos{rdx
= n Io"" s[n+t + cosrr
tlx
G*1)'.i
f':t tk \.tt* J,,, tir!;;;ot\ ) -
f"'. (1 + ctg2x)d(ctgx)\* )",r- ,rn..* * t 1
dr
, =- [o'"
=,([
: -(1",.
nt dx
sin{r + cosll(t + tg2 r)d(tgx)
tg{.r + 1
269
rl 1
lz*rzlr s rtlr+ + zJz
Burada tgx : t avezlamasini aparsaq:1
i=81'+#dr=8['=dt =8[
8 r; - llr= _.=nrct{t _...._lo = ,r rr.
1275. ('-/r = fo
(rsirrx)rdx.
Ecltl, I, =.[o'(xcosr)'dx inteqralml gofrrak va bu int€qrallan tapaq:
Ir? tn x3l" ,I.3
fr- I::.f {r"ir" + x2cos:r)dx =Jo,'dt = =,l" =;.rn
I2- I, = J (xtcos'x - x2sin2x)dx =
I x?=tt. Zxdx = du I
= I xrcas?.v6x = I 1 l=.lo
-. ----"---- lr,= -srnZx, dr'-cos2,r I
= 1 r,,, *,1','- (,u^2, a r =
= l; =:; =
;i::::;.1= _ [(_]-",.;t. *]l'-,,.,.1 =
/ n 1 '\ iI= _,._Z + yinzxt,)=1.
1,,-,,=+ - ['==*.;[-,,-,,=i |.,, =+-:
Belelikla, 3r(xsinx)2dx -- 6 -i
, (,- 1)
(' - i)'* ,
l,
270
'fo ("o")td' =*'i
1276
I = [" "' ror'= rtrr.-,0
IIaI& Sada gevirmedao solra yazarrq:
t =l [' "'tr+cos?t)rlr =1 f'"',t* +] [""'.rrz,a. =1,, * ],"ajo 'Jo -Jr)r- tt
I,=le'dx=s'l[=e"-r.,Io
i, : .{ e lcos?xdx inteqrahm hesablamaq 090n awelca
J' e^ cos?xtlx qeyri-mlrayyen inteqralmr trpaq:. I p' = u.dr, = coszxtlx I
I "'roszxdx =l I I =I -'*-^-- le'dx=du,r=-sin?xl
--)"r,i,,2, -|[ e'sir',xdx= l,;; =:::, =:i::,1 =
1 I lf=
--e'sinzx +;e'cos2r - nle'cos?xtlx.Buradan adi tatrlik qa1'dasr il: alaltq:
' '-(2e'sin?x ' e'cos2x\'I e'cos2-rdx = -,
OndaItt I tn II e'cos2xttx = 172p'-sin2x + e*cosZx)lo = ;(e" - t)Jo )
alarro. Belalikla verilen inteqral a'a$dakr kimi hesablanar:
t='(e,cos2xttx =lir + 1r2 =it." - r)+;(e" -tH(e" - rt
1277. r, ='1rin' *0
E?ltL lintf{ltsh hesablamaq 09tln hissa-hisss inteqrallama dusturunu
totbiq edak:
271
u = llrnn-r x,dt = (n- l)sin'-2 rwrd,dy = sin xd = y = - coc x,t
.tir, =-rio'-r,*'1tr *b-ilra*z ,w2 * =(,-r)jsi^"-2 ,$- rin2 ,fu, =
u,,,
=(n -t)lainn-z xe - (,r - r)JUo ".ra00
Buradetn = b-t}.n_2-Qr-l\n
alaflq. Ag3r sad.lofdir$k
\=*1,-,olrr. Belalikle, I, -inteqrslE hesablrmaq uglh r€kulent d0shr alrq.Bu[8d.a n=2b goalrsC<, alarq:
, (z*-rlz*-t\.....t.r, (z*-rlr ",2k = --;2k(2k
_:k(2k _.2:,0 = -@ll}i-'zBuradan
ai_Io = i6inrod=+6t
a=2t+l g0tlkak:, 2k(21-2).....2 (2lll,2' *r =Gr;JXr:l);:JJ,r =@;fi
I;t
Ir = Isiord=-cosd; =l0
t;127& t, = jwn *
0
flcn*,,=,r(?-,),.[ni]
oldu$mu nazam alsaq, verilan inteqrall,u
, - = 1
"^4 t -,\* =E -, =,1 = 1 "^" *",, \2 ) 12 16
272
alanq. Bwada Sqlrlan swelki inteqrahn hellinde istifrda etdiyimizqaydadan istiftdo etsek alanq:
i " . lW; n=2k,daqda16l .vlr={ - .
'o I l2h ll" ttz;;D ' n = 2k +r otdnqdq
,_
1279. 1, = lryz" *a,0
IIalIi, Sade gevirma ile
tg2n ,& = tg2,-2 t. lrz * = ,ru-2 r{la, = tt2*2 r.1-$}* =
= 82"-z rd((*1-,tzn-z *kimi yazanq. Axnncr ifideni
[o,I] fareasnOa inteUraUaV4:
,trit, =lqz"-z*1nny-o1,rr,u* ='{!-\l -r,-, =, -1,-,
0 0 '" 'loBu qaydaru t,-,,t,-r,...-lsro t*biq etsek vs
u=1a. =i;4
oldufunu rez:ra alsaq, alanq:, I , I I ., I I I,,= 211-,"-r=ij-7;tt,-2= 2n_i- 2n_3+ 2ll-tr-3 =,..=
I I t (-lP'-(r-r) Gi2'-('-z) (-l)2'-('-r)
-+
2n-t 2n-3 2n-5 2n-\2k -tl 2n -(2a -3) Zn-(zn-t\'
'(-rrr^ =(-rvlr-[r-I*l- * (-r)'-' -C.!I'')'l' '14 [ 3 5 2t-3 2,,-t))
r2ko. 1, =ii-,,Y *0
Halli
avezlemasi apnrsaq, r = sin I
ih = cxlstdt, O r, ,;
oldulundan
t,,, =ift - sl * =i*,2'*' ,a
00irteqralrx alanq. Onda
t
r, =j*"*'**=,9+-*++o (2r+ llll (2r+ l)!
l2EL t,=l;-eor'/l-:'
HalE Burada r = sin r ovadcmssin apemq,
ax=qstdt,O<t<I
oldulundan
ltr,=1La=isir,"ao"'/l-r' o
oldulunu alanq. Bunu ise )ro<andalo misallar kimi hall stmek hanndr',
,rrr. ,. -1r1'n,--" '1,"'t *'' 6\sltrr+cos-r/
IIallL Olava bucaq daxil €finoklo,J1 r lsirr-cosr=J:[rsinr- rlcc
rJ =
= Jr($!'.,s; -cos,*;) = o",(; ")
,i"'**,, = o(f ,o,,*f'"-,)=
= Jz[.^,*, in*,- ;) = O.*(X -,)atanq. Bu ifrdalori verilen rnteqralda nczsro alsaq:
,t
r, = (- r1t" t l,rz'. (, - IY = -i,r*., r,
atanq. Burada f-r=r olduEu nszare alnmrldrr. Burada
t, =-!+t,-1
harada ki, naticeda
r, =GI)'i-r,',D *1[r-].1- .crf-'ll" "L 21 2 4 , ))
rekurent dDshrunu a - t defc tetbiq etnekla
,- = L!a' * Llf-' * (- r)2'-3 * .. *" 2n 2!, -2 2n-3.#r* *ffi*(-')".'ro
I
- a o.dnro =
i,cd - iffi = r,"*4) = r"r:
il
ahnq.
*.r=f['*"-'] ,r rir,r=-][" -"-'l Eykr dwn anndan istifado,2edarak asattdah inteqral lan hesab laym:
f
12&1. jsaz' ,.u2" at (n ven ndral adedlerdir.).0
HalE !
t\2n,2n) = j",n2n ,*r2' ,,1,0
kimi ipre cdak. Hesablama UCii,n hissa-hisse inteqrallama dusfirunu tetbiqedak:
r =:in2'rccs2'-1..gifinrssk, ooda
dr = 2m si 2' -t x,;os 2', - 12, - l)sin 2" rcas2n - 2 rsin.rlr =
= [,, sin z'- r rc* z': - (2, - l)dn2'+l ,*2"-z ,lfdr = cos xdt = v = sin r
almar. Odr kr,
l(zn,u) = svr2'+t "*""-' 4? -
-lpr"n" **"2'x-(zn-t)sir3^+2 xo*zn-z .b -0
tt2--2
= -2ujsinz' xw2' xe + (zn - t)jen2n+2 ,*2n-2 gt
LQn,b) + 2ntQn,2r1 = 12, - rl,f "a2'
* 2 r oo.zn - 2 *0
Buradanu
t(z^,z,D= ffil"in2 +2 rc*2n-2 t&
ve ya
r(z^,zi = ff )tQn + 2,2tr - 2)
alanq. Prosesi z -t defe davam etdirssl swslki misallatda oldulu kimi
,e,.d = 6;.%Hffi*:y(2n + h.o) =
_ (zz-t)tt<zn +2n-t)tt t=tMr=r$n)r(zm)t - 4zn)r(zz,\r
2^+'+l(n + n)lzt+n mtd 22a+2h+lml (n+n)t
12u. iY/. &.0 $nr
IIIIE llrrrydalu /(r)=H firnksiyasr r=o,x=tt n6qtelodndan
bagqa her yerds byin olunmug funksiyadu. Burada Eyler dtistsmndan
"a*=jb,* -;,*),,", = j-G" -,-')
istifide etselq alanq:
216
gf ="'^ -"-',* = i"iKn*r)-ztb =sinx "n -e-E E,
[z["or(, - t)r * "*(" - t)r + ... + cos r] n to& olduqda-
lZ[cos(n -t[+cos(z-3)x+... +cosr] a ciit otduqda
*1'*l(n + r) - xfi* = o (k = t2,..,n)
0
oldu[undaq alrnmrg ifrdeni ineqrallasaq, alanq:
fsinra*- [0,, ci.it oldrqdr
I siar [r, n tok otduqda
12g5. ictr,(2a+rb *.i*,I/alli Eyler dflshrundan istifrda edek:
*r(zo +tY = !{O4u+I! * "-(z'+r})*u.. = ]['*,-o)*"(2, +l\ - "42o*rh
+ "-42"t)'Cq* .o + r-E
= "t2ar - 4(zz-z\t. *...+(-tl +...+"-i2* =
= z E(- tf -' "* z[" -(t - t)} +(- t/h1
Axrnacr ifadani [o,r]-de ideqrdhsaq, alaflqir = (_rf r.
BuradatI@s2h-{r-t)he=0, t=G0
oldufu nezere ahnmt g<tr.
1286. lwal.cxr" xdt .
0
Haltl
*t',- '
*"" ,"n"* = #Qi, * "-*ftm * "',*)-= i" if :b'o'-t\ * "-'znl=
P:In'*f== #l' * .
ili," I
"' 4' - tb + r *'21,c r
"-'' af
=
= I * I l5,cf
",4,-ry ,\t
"r"-iz*l_Z, zn*tLEr " A ,, J
= ] * ] "j'cf "* 4, -r1,2' z"Eo "
Arrnncr i&dani fu teqrallasaq,
lcuz(n-*)*=o0
?*'."^rort=ai2n
lnteqrullan taptn n ,uluml dddir;1287. Js;nn-r x cot(n +t\x,lx
0
.IIaIIi lnteqratalu ifadds tcplama teoremindon isti&ds edek:
rin"-l rcoo (z + l)r = sir'-1 r(cos r* croo r - sin zx sin r) =
= sin'-l rms nr cos r - ria' .rsiunrBu hatda
,,I = Jsin'-t rcos(r + t)tat = Isin'-l rom nrcos *dt -
- lsin" nhatb0
Birirrci inteqralr hisss-hisss ideqrallayaq:
=fi1;"*""a-rv '
olduf,undan
almar.
27E
I1 : f "i.'-l ,* .o.ou rd =0
f, = "*- y'v=sint-lrco:xd
I
=
l* = -^^ ** v = Jsh'-r rdGi! r)= i't' I =
= lrir' ,-" rl' * I.'r' rrio *6 = f ,i, n rsin rt&a lo6 o
11 = Jsin" rsio m&0
Bu ifrdani l-inEqElmda mzaF shq:tt
1= Jsin' rsin rcd - Jsio' r aia ru& = 0.00
tI2t& jcos'-I rsir(z + lld
u
IIaId, hrqnhlt funh iyau./(x) = coe'-l r sin(, + 1! = c*z-l ,1tt - ooc r + cos rr sia x)
yaaaq, veribn inEqnh
I =tlols'i xsin ra& +fcos'-t , coo rcsin rd00
kini yazanq. ftinci tophmm hbsadissa htsqnlhyaq:t-
l2 = Jcos'-t xcor zrsin xd =0
L=*- dv = q'-l rcin rd It-l=
la, = -o"i^ *, = -lcosn-l rd(oos r)= - | *' J
-f'.'nl
= - I o*n r* -l' - f cc' rsir ,ra = -fcos' rsin rad, lo 6 o
ahrq. Axrrmcr ifrdeni veribn intqralda nezan ahq:,rt,
I = jcoo'-l :sia(n+ tla = Icoc'rsia ryd + Jcos'-l rcc nr sin rd =000
tl= Jcos'rsin rr& - Jcosz.rsin rcd = 0
279
2tr1289. l= ! e-G wzn x&
0Ealll Burlx- fubr dGtrrundan btifada edek,
"*, =l (.o *"-o\
"-^ *"'" , =ffliz* *rtr
"42"-zb * c2u"il?n-a\ +...+
+ c2, n + Q1- Q"- t)
"-,12, - zb * ; tul=
ff ,
+ c!, cu(zn - z*) + ... + c l;t co,x + c l,\,frQ|, *z * * c1, ,,s(z n - 2\ + ... +
2tI cos(zn-zt)xe-**=xe0
xl(2n - zt)slm(2a - 2k\ - a @s{2n - 2*bl," =
= !;u'*1,, h=0,t2....,n_lo2 +(zn -ztf '
oldugundan, t -intsqnLnr agagdakl kimi ahnq:
.{-a*t(za-t*))l2t= Re :----------J
- a+ iQn-2*)lo
2i e4-,+i\,-21\)e=
0
e-'E
o2 +(2"-ztf
'=$l("r'.';i,'x7fi:a)I
1290. t(n,n)= l,a-1X- rl-16 Eyhr intsqotnr hbssibsa0
intsqalhma dlbturunu bir nega da6 b6Q edeok hesabhym, bunda nvo z natrnl sdedbdir.
HeM, =1t- rf-t, dv = x'-t &
g6fllok. Onda
tu = (n - rft - xl-2, v = ! y^-r 6 = L y,'
oHufundan,
280
a (n, n\ = L'^ 11 -'f -' li . i i,' 1r -,Y - 2 a = !1 sb, * rn - rr.
a(- + r., - r) =j x' 0 -,Y-2 *inteqralna hisss-hisse imeqrall,anranr tetbiq edsk:
, =1-,f-2, d, = -(n - 2fi - x)''3 d"'
dr, = rh &, u =;firt*'
B(n + r.n - r) = )-,-*r 1r -,Y-2lr +
* n 1|r",*rtr -r), -3a=!:10(^+z,n-z\z+l;
Yenidan aldrf,rmz axnncr ifrdsyc hisse-hisso im€qra[amafl tokari tttbiqetrrakla
u (^,,) = 9#i##)a(zr + r - r,r) =
= ("-r)l \ rn+ n-27x
=n(n +t).....1n+ n - 2li1
(" - t)ri6+r)---(-+"-F.+n-t)
Qahyular:E" _
1291. Hxe.-lE
1292 Jsin&.0
Ji a'l_12e3. I r*r?
6sh2 dx
1294. I =:;cirl {l+r"
Cavab: 1114
Cavab:2.
Cavab:1.6
Carab: L
2tt
Int ilr-r4.02t
t2g6 ! ___2_ (0<e<t)O I + 6CO€X
1299. fim I=l
Cavab: l.
cauob, -4-.,lt- 12M0alyan htteqrullann hhmayi ib limitl*i hesobloyn
,.-- lP +2P +...+ nP Canab, Irn7. ,s----;;;;- *'"" p*t'(z>o)
Yltksah tofiibli sotwuz *ifildrrbdn otn aqla limillari haoblayn-. tt n I
r2sE ,'Y*"^; ;i,;;q cavab: frn
(r>O). Avab: x+l,,I
(!.Iz,l30ll lm I ----; +
,-+o[n + I
Hesablaytn
I3ot. llra*'*(&a
,23 r)
1ftc"|=r..=rln+- n+- |2 n)
Cavab: -Lln2
1302
130s.
1304
1305.
dd"
ddb
dad
b^Jsinx'&ob"I sin x"drq
x2
a
x] dt
Cavab: 0
Cavab: - sin a2 .
Cavab: sinD2 .
-t2dtCavab:zx,llJ .
al,wd @sx ,1306 . I cos(rt')dtd sinx
,-2 .-Lbvab: _--_-
Jt+"12 rh*rECavab: (sin x - cos .r) . cos(z sin
2 r) .
2t2
(m+k\(nx+k+1)
l*"12d,litu o
.r+0 x
1@",ei2a13N. th L-_=:-
r+<D r/.t, + I
Crnt,44
Cavab: l.
Cavab: 0.
carab:1.6
ICavab:
u16
Cavtb:2*L2
130. [mx+@
ft"r*)'[o)-*,Ie'^ dr0
HbseJl,bse h*gabm balutun bdfib eba*h*qolbtlaab@ut
h2 1o1310, -!- rg-x 6 Car'ab: i lai.
o 22x Cavab: E.
1311. J.rsin:dr.02n . Cavab:4r.
1312, I r'cosx&0
e c"*u' z{r-l).
cr-t,4-{.32
rlrs. iZ-1^15 - 4xa . f-;----;
1316 Ix',,1a'-x'&
lt2 r-1317. ! ,,!e' -r,lx
2t1
rLtr. {h'tr';J1
1311. Ixortgxb0
Wvtd.wzcam Buh b mbyym l*qdbt laobfon
,rbwn
131t.
I3Is. l4r-,'1"*0
ri2o. 1@ui2a,I9 ^_
1321. !rrJl- x dxI-l dx
1322. I --.-2 xtl x' -l
03
1321. Jarcsin02n
Cavab:315!.26
crncb, ar3 -L27 27
Cavab: -669.7
Cavab: -43
c""ubr 29.270
A.Carab: -a - J3 .
Cavtb: 2/,.!1.
corb,lbz-225 .I 1024
Cavab:1r=#F
gsYa6i !-sra!1.2n2
Co*Art4
h*gallot lvsablayn
7-t*Il+.r
&
Eybr dbtvfut rdan bdfrb cbn* lrrE$ollut lcsabbyt*
l*gatUt luoUoyrn (n-rpt ralobtWJ)
2U
1323. i*"Jiu'*
B2s. I (2 + cos x)(3 + cos.r)
1 c"*u' I1326' lsin:sin 2.rsin3r& 6
0ln2
1327. I sh+x&0
n tumabdowt qda ddbi aytt sohtu d,trttra lb npoLrrsjirdanafu l*qolbr lsafuyr*
IJ2& tn=lr'qr4nar0
1329. isin' xsnm&0
i cavab:4(-l)'-1.I33o- t1,sr.coszmh 4n' '0
s3. oRTA QTYTUAT TEOREid
1. Fut*siyann orta qiry"ti. fa,tl pogasuda 7Q) funsiy'ostnn ortaqiynati
"ut=*it',w*W'ial
funtsiyast t,rl i*rr* bsitmaz otarso, otda ela
c e (a,t) niqtasi var ki,ultl= rG)
2. (hb qrtmt hqda bthrt oocm Agar: 1) 1Q) ua e(r)funlaiylan fa,tl pargannda mawud va inteqrallanan funlaiwlardtrsa;2) dx) funfs$Nt (s,b) aruhfrnda itoasini dayismirsa, onda
bbItGhFta.=pls6t*
barabar I iy i dopzdw, burdanSpSM
ua .= nr 1Q\ u= "q lG)a<x<b a<x<b3) 7Q) fwsiyas fa,bl pargasrnda bsilmazdirsa, ondo
t=IG\olar, brtada o<c <b.
3. O?n qbnct hoqda nbd corum OEar: 1) 7Q) w e(xlfunbialan la,t] pmgasmda maMud ya inteqrallanon furcialordrsa; 2)
9lr) fanhiyast (o,b) aral$mda monotondursa, ondab6bI fG)d,W=oG*o)l fvb+e<b-o)l fF)e
barabarliyi dofirudr, bwda o<{ <b; ,. dx) funb@t mar{i olnqorua monoton azalon funbidrsa, ondo
00tkimi gdsaak. Burada
evezbnrxhi apasaq. onda E=t +t
r le=d lo nJrsin;a = lr = I qlduqds r =-zl= j(z+r)sinQr + tle = -!taintd;;o lr=ro,tqdar=o I -' 0
E=r+t I
2* l---*' I " tlzsin.x&= l* - * l= I rsit1,z. +rV = -r tslntdL F="dd4ds'=o 16 6
b=2xowqdzt=xlahrr. Bur&d8n
2rtlr
beI tEb{*)a = oG * o\ t6b, Q s{ <b)
a). fl,x\ funksiast manfi olmayan ua monoton artandvsa, orrdabbI tGh6b = q4b - o)l tG\*. (o < € < o\oc
Agafidah maayyan irueqrallarrn isarasini tayin edin:2,
1331. I,stu th0
flald, Veribn intsqnlr2rr2t{xsimx&= J.rsin;& + Jrsb.rd
Jrsiar&=-Jrsinrd -t lsd Ut00i
rEtbe ohEqtJsin.rd =zsinf >0, (o. 1. n)0
ohufmdan br iki ophmn menfdir. Odur ki,2utlxtinx& = -rlsnr& <000
2U
1332.
boyllkd0l?
EU
I2
a) t' = Jsinr0.ra va yaxud0
[,,]]*o**"
t2
t2 = lsnz r& ineqnlhnndan ham r
0
sio2r>sinlor,
Arcaq r= o,r = I qiFretbrin& bunhr be ube rdir.
sia2 r -sinl0.r > o = sin2 r > sinl0.r, *. [0,1].. L 2J
Odur ki, r2-intsqBh ll-inteqnlndan boyUkd0r.tl
1333. b) y=je-x& va yaxud t2=!c-" & btsqalhnndan hars r
00boy[kdur?
II2A, e [o;)- at "-" r"-'
odurki,
olur.
Iz>lt
1i31, s)fb)= x2 firnls iyae nm [o,t] parpas nda orta qiymtini bpm.Ilzll Oru qimet tsoFmb gdF
uUl= J-ol a,b= +i,,* = :,,1' = I"' b-ol"' l-oi, , lo 3
1335, b)lG)=J; firnbiyasnn [o,too] pepasnda orb qiymatini
hPm.Halll Oru qrynot Eoemini btbh edak:
"vt=*itw=#-'[r*=;100
2lmm100 13
0
1336 cI /(r)=lo+zsinr+3cos.x ftnk iyas nm [o,zz] parymm& orte
qiynatini upn.EA
2n
, Vl= *i l,l* = ia'f <ro,, * x + t cos t)& =
= |(to' - z *", * 3 ri^ *\'o' = * (20 n - z + zl =ff = rc
' l- 6 cwpqiyrctini lapm.
EA ('| _l
" H = *' [ #k = :i#u = +li#k. i#t' , ')
Buada intqnhltr funh iyada mueyyan gevirmo apanq:
| - a as s = sss2 9 a r*2 | - " *"2
$ *
" "a2 | -- {r - ")*r2 f+ (l + s)sin 2
;Bu ifidani axrnncr benbedikda nozen ahq:
*,,r=rli * .i o, 1=-'' - ;
ll V d*"' |. g + e)sn2 | 11, - "1*' q * 1. ")'t' ;.1
=-r-ll 4.1 -i 4"',) 1==;0;al[A.,'*r- f;ay= #rl# --G t\: . E -".c f );
]
=
I - 'o -,11 2p Jtri, p= l rl:.Hrl.)=#l#+=#
Amlitik bndasodan.[7:F b2
"= o 'P=;oHulu melumdur. Bumda a$6y0k yarm ox, D-be kigk yarrnoxdur.Burhn alnran notic"* *oo oblirr=
,
oHugu almar.,
I3J8. thtaq ki, I tOt* = t(a) (r)0
Aqag&kt fmlsiyahr ugltnd{r bpm:t) 71r1=tn, 1r, -17.
b) /(r) = ln, .
v) 11t1= s' .
lim , Ye Iim a limitbri naye benbediflr+0 ,-++@
Ealll t)
Itn& = r(&)n0
.n+l lr n+1r"-'l -tara+t + !I! = 6n r'.+t +0o = | =e=
|,+llo n+l z+l 4n+l
liod= lim e = --]-=.t--to r-tro 4n+lb)
lin 0= tin a=!r-+0 t-+-l<o e
v) Jr(r)=r'{i (1) beoberliyinde yerioa yazaq:I
Jet d = xe&0
Buadan
"'-l=o*almar. Bumdan
289
i^,* ='^*0
, ls=Itr1. dv=e\ r
!"* =V = +,,=, l=,u4
-1a =xhr-r=r(rnx-r)
r(ln r - t)= x(6P16r;lnr-l=hd+lnxltl=-t=0=e-r =!
e
ee =t:lue 4sas ra g6re loqarifinahsaq, ahnq:
a=,,{-lT
ve ya
s=t k'9::) "qtx- ohun. Onda
b1:1=;.d{)oHutmdan
lr,,=lsi(;.4')=iahot .x -+ +o obun' Onda
lim d=ltr'16
Birtrci orta qiqat teorc dan ktifade edank oSa{@akt
int e q rall an q iyn a t ldndirin i
1339. t=zf b; I + 0,5cs x
IbUI B irinci oftr qiyrmt tsorcmins oossen yazsnq:,i * =ri t = 2o -!r<t<4xI t+o,5cosx -lt+oJcosr l+0.5cBf 3
qlmotbnd ilma do!rudur. B undan
-!o.t-!n.!,333r-!,3 =e
4-t3
iao e&k. Onda
, =i"?= it"("' -r)-h,F r[h,'(,-])-',] =
= |['.n[--')- ",1= ]L -,-'.4')-t,l
290
tl**!*e, @'r)t_9
1310. t=I->diJl+r
flrrn B irirci o a qiymtt teoPmino g6D
v--LloJl +6
buuds o<6<l ondsl-.t.1
t0J2 l0rhrq.
I _,I3rr.8) lirtr li-*=o oldu$unu bbatedin'
'-+''6 t + .x
IbbOtu qlYrcttsoGmino goE
i*-=o#o' (o'r'r)
beblftb
,!-.i*"=,goth='
@hgrlr,iol,:
Brtrct cn qlytot tacr&bn tdfib ebrah bxqdbrlsob@n
loo .r Carab: 0,01-Ofi)54 (0<A<l)1312'
{r.t**l*-tt*t ot qtrttct amct tfu bdlstb cbt* htgaltt la oblayn'
2@r sin.r .1313. I -dl00z I
lhu. bre- tu ,' xe @>0,atb^
1315. lstnxz& (0<a<b)'o
cavab: d (o<d<l).
50t
0<a<D) cavab: Zea
Cgrab:9a
(14. t)
(ldl. tI
291
51. eE nl4[ a{$rsi ht uguuz nl. Biritrci n0v qcyri-mersuri lnteqrelOgory : ,f(r) funreiyss r a S .r < +co araltmda tayin olunub ve her
birsonhr n < x < b < 1oa pryasrnda intsqnlhnendrng bri6 gOe
[*' f(*)or= tim I f(x)arJd *-. -la
(1)
dotrudur. Bu inbqral bttst ,thv qqrhms rei la:Eqal ad1Elrlr.. Ogarx - *m oEuqda f (r) = f /(x)ax-n sonlu timiti vasa, onda
l'i" f G)ax intsqnh y{rlat, aks haHa be in6qral da!tun adhmr.l,l-Kogt blElct
V€ > 0 fr > a varki, x' > x, x" > r qiyrctbrinde
rF(;r) - F(x")t =ll'," tur* <tgertinin ddenmesi (1) ineqnlmur ylf,rlan 6lmac1 {lglhr zeruri ve kafgsrdir.
I.2. IQcyba abnod.Oger (a, +cn) aalrtmda iki inbqmlhnan /r (x) va /. (x) firnfsiyahnttgtln /,(r) S cfzk) (c > 0) 6denese, /r(r) fimkciyasrnm inbqnhnmy{tlmasndan /r(x) fimbiyasnm intqn.Lnm y{rlmeqr 9orr. /r(x)finlsiyasnm intsqnlmm dafthasmdan /r(x) firnkiasnm intsqalmmdafirlmas 13fu1,9O.
l"**lfG)la, .,)inleqnlr yr[&ndnsa, onda (1) inEqra h nUbq ygfun begal adbna.
1.j. @yri+cs re t hUqah ygrhrct fufrn Dt&;h elatmd.
[" f {io{*1a* (r)inEqmlms taxaq. Ogsr x * o olduqda kos ilmez diEremblhnang(x) funls iyasr srfia besber ;.(x) furlsiyasmn rnehdud ibtidaifimh iyas dma, onda p) inbqrah y{rtrr.
2. furci n0v qeyrl-marsusl inteqnlOgar /(x) nrnts iyas I [a + e, b]pa&asn& mehdud ve inbqnflanen
funh iyadrsa, baqqa s0zb bofln [o, DJ pa4as mda inEqnllanan deyibe,yazary:
[*- lkror = lirn F(e) =lo E-+ 0lim I
s-+o .ln *"f(x)dx (+)
2E2
Bu limit vasa, (4) ftinci nov qgyri-rra>srsi incqoh yrgt4 eh halda bod"E fir.
Qeyd edak ki t = * avezbmes i ib ikinci nOv qgyriaasrsihtqah bidnci n0v qeyri-oasrs i inbqnh gewilir.
1316 hifu,k0meyi ib ineqnt adqiq edin:
[o nr..l__l + x2
tuafo dx io dr/ -i+ =,lil" I 1ft =.lit-o'ttelE: -lim-(arcteo- arctsg =!
intqnl y{ilr.IJJ7. Hesabhym:
,_l'- dx'-)';':+'x-Z
EA
, = i,' -'
7{1= "!* [' #r+ =,r. i, l=;*ll,: i,ti*" l-ll, = igl,- [', l#l -,,1J =
I D-l I I 2:5olim ln4;;1 = -ln4 = -1n22 =:1n2.intqal y{ilr.
I3J& Hesabhym:l+-x2'lJ. *\ td'
H.U.
t'*'t2 + t rul+* rr ri(x-,l)l" ir-rdx =,riT*1. ;i!o- =,,iTLl. ;-fr=
D3
r _y:_ llb 1 I b;_l _tl= rlif.;n,.te ,.-r, l"
= Jrlit]t. [nt.t., -Ttnrct, 0-l =
Lfi JTr II=-{-+-l=-_,1tz 2t ,t2
1i19.H*abhYm:It d.v
, _ 1" {z _x)r,T=T
Eattl I - x : t: evazbrnasini apaaq. Onda dx = -Ztdr ohr'
,v= 1-t2, Z-x=2 -1+tJ = 1+t2, x =0, t=1; x= 1, t:0.Brmhn ineqmlda nezan ahaq:
t = JJ," a+c= - l:"#+ = -zn,ctstl :l
I35O HesabhYm:
,=1," dx
.YVl+I;:FHcltl Owehe qyri<ntlayyan inEqralt bpaq:
I d, = r ,i, = _l r:jGI_=I ;r, - | ,..,,SF t',,:(+)'+(])+
r
= 1}=,it.,isn,,"a,kl = :i+= lf -$=r1(t+2)'+4
1. | 1 _=-=l r lr , ,t- ,.rl== -:t,, lr*l*,,r:- ,- rl = --t,, lF.r* *,-*F*'l
2xs- -t^-3"'z * xs + zrTJli Td
)94
2
(
l-x1
1m
2
li
II = lirn :In,-+:( 5
2xs
t:!1*r-+tJo-2b5
b:T; s "'\2 + [s + 2llllill]a2\
- lrr ---------= l:3 + Zr 3 /It 2 ? r 1 3+2v3
= -ll11 - - ln--------:l= -ln-.5\ 3 3*2\2t 5 2
I35I. Hesabhym:
'=lo (r-':;; dx
Ilalll Owebe qevrirntlayyen inEqal Apeq:
i xlnx lrr = tnr,dr, = ]{r + rr)-ratr * rr)lI tt-r*o-:1 . t.' r I l:' I drr=-d.r.t':-2'1*i
I1 ln.r 7[ dx
2l-r: 2.1 .Y(l + x:)I a., .4 8r*C ,l(1+r:)-8rr+Cr F+B=o.l;G;f = i' I *.' =
'(1- .) i t, =?I lrrr lr f tlx i r' \
= -rt* x: - l(l; - J ;;o')-l'l r I
I lnx li 1 r.
= -I r - r' * 7(lrrlxl - ;lrrl1 -'rrl) =
llnxlx2= -2t- rr+iln r- *,
- lnx 1 r: lo/ = Iirn (--'---------=- * - In -'--------- ) ID---o''2(l .x') 4 l+xz'1.
I lnt I r: \=lgli-zC;l--t,.'*")=, lnt I 1 I ,
= lirn [- ....1........... -]lnr- In I=s-c \ 3(1-r?) 2 I l+-t2tI r:lnt 1 I \
=.It'l,- r(r. d) + otn r - u)= o
Bumda lim e 2lne = 0 olduEu nazan alnmgdrr.
s-rO
iAda6hi
y86oq, buEdsn
13s2. t =*f ..-"vj d,o (t*r'F
frelll ar6r = y evozbm*ini aFBrEq, ifi&ni ditrcnsbllrsaq,ahnq:
. [.r=0, Y=6aE-t----; = crY 1 zl+ x' l*=*' Y=,
gcEx = Y
B@"tg,\=ts'
x=tgalmar. Bunhn vcribn inbqalda nezan ahq:
,,;;-1 v .1 lY=u, dv=cmYctYlt= l---dY= llco.vav=r r='orh*tg2v' it' "' 14=6" v=sin'Y I
L: f= ysin 1162 - j-*" W = i* "* n& = i-r0-
Daaeni ozaltma dlstwurun lctmeyi ila ogafuah qeyri-mexswiinteqrallot lresablayn. n aalural adaddb:
1353. tn = [r'e-'&0
.[Ia&, Hbse{bsa inbqtslhma d&trrunu btbiq ed.k. Onda ahnq:L= r'- dv =e-'dl, =l 1=-r's-tl*'+rf r''re-'& = nt,-tlfu=m'-'&, v=-e-' | 'o 6
I,-1-inteqalrne hbsedbso inbqEllentr d[stuunu blb{ e&k:
4-1=lx,-te-,&=l=*-t' ^ ^=n'4='*' L- - - bt=Q-r)nn-26' '=-n-t I
= - r,-1 "-,1+'
+(n-tfi,"-2 "-'
6 = (r-tl,_zlo ' 'oBeblikb,
In=zI;-1 = '(''llr-Z
296
4=ie-'a=-"-'l* =r.olo
Bu Sydam davam etiirak, ahq:t, = nG - rXn - z)' -..' z' t = at
=l{,*!1'**-t'1"L\ a) " )
b u-b2r+: =r,
-=
toa
ib iaa edek Onda ahnq:
,,=*i =-4-.!.Qz *ry'Bu Gdadan btifide edarok:
lr-t= P d1.6;.P
yazaq. hdi hbsadbsa inEqolhrne dBturunu tekarblb{ e&k:
. -[,=6'**I'.', *=y'-"-' -
l*=P,*rlr2 *rl" zaa,'=' I
=*,ii6;i,-4"-,,r.f ffi== X, - tff ---!-- - 4n -t)a, =
_o\aa +k)
= 4n - r\,-r -fu -t)u, = z(n- t[t,-1 - u,)
"--r'.'\ffi=
Buadan
t' = #? it'-' = #ffiu, = =ffi. # =
(2t' - 3)t t T e-O;:it -
t^-troo
bn-t\t G Ie= *":lrfrlurlftttlfl
r(?t - 3llt a"-t sgd a
ln
4=i"a'a0
" =_&=.r\!-_- a*"-t(zn-z}tt
298
oHulundan, yekun ohnq ahnq:
a"-r12,-z'1t(rc-*l-i,rrr.t,=irffiHeA
,=""&'" iJ:7kimi yaznnqb, r=sin, evezbm iniapanaq, ahnq:
&=@stdr=0 olduqda r=0
r=l oldro& r=1'2
1356, \:f #,IIaI&, Hkso{bsa intqtslhma dBarrun b6[ edak:
l-j-\ u-02 -
L=-J-=r1r'o*rr. ^=4I ch'-Lx chz tllau=(-n+tln-'xsna, v= s I
,,=fl,'o* -?,uff 4+*==b.ff#-G-,ff_e-.'*==(n-rl,-2-Q-r\,tn=dt,-z
n
, = (, - tXa - r)'.... z ,.
"(n-z\.....t ''
,'=f 4 =,*,;=,O ch'x
Buodan
oHulun&n ,, -tok olduqda
alnr.
, -(a-t)l'" - nll
r+[t olduqda ba
r=q#h
^=i*=4r:v=;t#=b'=,, .r=0, r=rl
=lP**=a, r= o, r =o{
= zi -4. =,*"wF = {; - I)=, i = ;| =L.Q-t)t't'n 2 |
lt
1357. i) tr = j h sin ra;0
t1
b) 12 = Jlncoerd.0
299
Ezl . Har fui intsqnl ikirci nov qgyri-mesui irbqnlhdr. r = o
oHuqda t1-ineqnh, r=f olaUda r2inEqEt rD)(sus iyyab nralik olur.
r=a-r gOfllsek Iz =Ir ohr. DoErudan &&=-4r
x=0 oldrqda r=lt daoa, r=o,=j
oat,=-lrr,qJ{+-tV=f^,r*
" \2 ) i,,
IE
r, + 12 = 2rr =N"?Y =i(rrin z.t- rnzlr =
,ft*a,= lbsitrl4.d - h2 ! e = -+triz+ltutuzxd. = -Luz + r
00*i Pr=t, o<t<d ,,
r= ltnsnzx&=l__t o l=*Jromra=o lG=-a' I to
(t ) r,1, * l,=:l Jhsinrd+ Itn sinta l--lnsnd =y"lo r I o(r)
,I In sin rd -inEqE Inda ,=t-z ewzbtleri apannaqbI,
tJnsina=t1t
oHulu almr. Yuxandakr netbe bri nezec abaq
zr1=4 '!62ahrq. Bumdan
tt = -+ht2=t2.,2lrteqrallarm y$ masm dqiq edin:
j.JSLT ,28tn
'4 - '2 +lEcAL
'J'# ='i [*. I * o' (, - rIF = uF - 4I' . i,l:-' . o' b'}.. qt =
=nr-l+"r+o'|,}) (q-sabitdn)
,1 o# =
^F - rl?., * 1 -!:t * 6' (rz)* s41 =
= -tn p-.t+ rz + o'(p') 1"r -'rriatl
l0r
7*;=i.4") '-.-olduqda intsqnl yrlrlr.
nsc.ti -Li Al,2 +rEaIl
t t, * o(,') ,-.,-,E=-.xJ
olduqda intsqal yrtdr.
,r*.1*6h,
Eall! t=0 olduqda -l firnls iyas r me)cus iyyeb malikdir. Odur kim.t
briS gOn
,=1g=,,"'[',"g* ' ''6r,, i1oJ 6 " ,l#l (a>o,a>o) o<r<2
anlrtmda
*=cr'.h=!*l*o'(''-olOnda
Bu axrnncr iki intqralm cemi
1 = h i - ](r + /)+ ",
* "2
* o,,2)* o' (y2\p2
oHu$rndan s ve / {un srfa yaxmlapmacrftha asft obaq, I-intsqBlmnlirriti yoxdur. Odr ki, veribn intsqal detftr.
1361, l re-te-'&0
tuaa+(o
1= ! {-r e-, & = | zp-t e-, &, + ! xP're''&--\+1200/,
o,O<a<+o partini Odsyen qeyd ohrnmr4 odaddir. r-r+tl oHuqda Il'intsqalmr tadqd e&k Bu haBa 'o-t"-' =6,1 ,L -l r-p.r olduqda,
\'t-P ) 'yen\ p>o olduqda I1-intsqnh yr{rlr. Ir-inbqnh ikinci nOv qpyd-
mrc rs i in&qaHr. r -+ +.o oHuq& e-' funk iyas r \ 1l , ,1- t/tfirnks iyasna mzenn stotlo azalr. Odur ki, p'nrn islrnibn mthhol
qiynatbrinda 12 intsqnh yU rr. Odur ki, I -ineqnh p > o oHuqda yt$lr'
1362,i,p nq | *t-xU
Eafi. nL=t evazbmcini apnq.rL=/ a, y=s-t = &=-e-td, a<ts0T.
Ovezbmaai veribn intsqaHa oezae ahq:
j"1rr\f a0
Axrrrcr intsqoh birinci va itinci qeyrioe:srsi intsqa.llam cemi kimigct F bibdk.,-+0 olduqda
"++ttp =o.f-!-.|.u-q )
"$,.tlp =olr-tlr -+ r-- olduqda
ohr. B urada a I den boy0k ixtiyari edsddir. Ogsrp+l>0
olduqda Iopital qaydasr ih gGennek ohrki,
,,. 'f'l =0.l-+'+a
'++11)Bcblikb, verilen inEqntn y{&n olnus r 040n - q < r, p+l>0iafibrinin6&nma i hzmdr. Baggsfzb, q>-l\ p +l>o gartleri odonnrolitir.
+o1i63. I r aG>ol
I t+r'flalU
x-r+0 .ro,ro" $=o'(r')r++.o ordr4da :1=r'({=l
l+ xn \t*n )Beblkb, egar z>-t, lr-n>t ohaa, inEqnl y{dtr.
,t*.\ q"tss 6 1a* o)0.!
H?A
r++o "ldrqd"
d=8'=o'f ]r)xo (.r'-r J
r-++@ olduqda g':!"="'{+)x' l. r' ,,1
Beblftb,egarn-l<l ll>1 ve ya I<n <2 oHuqda hcqaly{ilu.
,J5t. T b(t! x) e
0rItaIL h(t+ r)-x mlum oEibi beabertiyine goe
h(l + r)_ I
xn ,n-lyazarq. Buada z-l>0 v ya n>l oHuqda veribn qeyri<raxsrsi
inbqtsl ygilrr.
136.'i t'-d* d, (n >o)'g 2+xn
Ila[t r-+ +o olduqda
x' qctgr - "'f
r l2+r, 'l-5;1j '-*
oHuqda
303
,, -ctgr _ ar( t \2+ ri \r"-. /
Odur ki, -(n+t)<l ve n-m>l eyni zarnrnda O&nildikda, qeyd-me:rsrsiintqnl yrt rr Bagqsfzb, n>-2 ye n- m>l olduqda intsqol yrtlhr,
1367,T wu e, (,'>o\is l+ t'
Ealll. wu flrnksiasr mehdun # **r ie r>o oHuqda
moootan azahn olduf,undan Dirixb ahnntine gOE intqnl yrg rr.
,r*,*i str.2 x *Ir
trIaIJL Veribn ineqrah
T4"=foa a*j "b!*, * (o>o)'o x itx
kimi yazaq. Buada
isin2 x *.i*-4' =o'I r 6 2lo 2
oldulundan, onda bu intsqal var. -i
** inteqmlmr be
,r,,, , =i0 _ *, z,)
aynlry nm k0moyi ib*f$Ii.2xd _!i&_1f-rz,*' r 2Lr 2L x
kimi yazanq. a
. ^_ =
"l,lj* = -!;
yani, intsqral dalrlr. +i w2t * ineqnh be cos2x firnls iyas r mahdud, 16rr
ftnksiyas r be monobn azabn oHulundan Dirixb ehrratine gOle y{rlr.odur ki 'i sin 2 r d ineqot drEilr.
6.I
1M
l:,ec.lt---!-I sin P rcc{ r
E U -----l- fiub iyasr qalr ve yuxan safradde me)susiyyebsin P rcog r
malftdir. Odur ki, vedbn intsqEl ikinci nOv qgyrioasusi intsqnHr.r -+ 0 olduqda
' =o'f t)sinp rcu9 .r (rp J
.r -+ 1- o oHuoda2' ttI =o.l tlsbP.rsin,r L(;_l,l
kimi olur. Odur ki, p <1, q <t oHuqda yrtrlr.I a,
/J70. Ix\or/l -r'
EzB. Yerjbt htsqnl r = I oHuq& naxsrriyyeb malik olur. Yeniintsqnl ikinci nOv qgyri-masmi intsqnHr. r -+ r{ olduqda
#="(+).r -+ l-0 oHr.rqda be
r' ^.f , IG-"L(';lBeblikb, z>-l oHuqda inbqnl ygilr.
1371.*i -3-'o Jr3 *,
.[bE r-+0 olduqda
=,'[+][,, j
IE.r -+ +o oHuqda bs
305
"[7lIffi
ohr. ineqnl y{rln
tttz.*i *to x?.xqIIcAt Blur'& p<q, P>q halhnna baxaq.
dalL tt. p < q obun. Onda .x -+ +0 olduqda
.r --r +- oHuqda ba
= tio += trm ---# =,llilo[-,r),\'* - 2o- t-t
' =o'( 'IxP +x4 \xP i
r ^./t)
-=u
l-lrP +x9 l\rq J
ohr. Odurki,P<1, 4>l
olduqda ineqal y{thr.-Og;t prq ohea, onda yuxandakr halda oldufu kimi 4 <1, p>l
oHuqda ineqnl Yrfrlr' Her fti haHaurin (p, q) < I
nax(P'a)> t
oHuqda ineqnl yrldr.l.
Bn. | !]-6airl-r"
Ilalll I
-. ln.r -. ; Ilim __::::_ = Iirn :_j:_=_:
,-l-ol-x2 t-+l-00-21 2
(Lopital qaydas mr tetbiq etmakb)'
noqtas i atra linda inEqnhhr fimls iya mahduddur. 0 < x< I ohrxr. Onda
-. lnx I ., | ., ln.x , ,. lnxlim -'-.:-= lim ---------:- lim ------i= 1' Im -----i =x-++0t-rl yl t-++'J1- 1L r_+'01-^ r++0.r_a
(hrI ri.4=ot-+tQ - A
p=q olduqda'intqel
306
Derneli ineqnhlt In'- frnbiyasr .t=0 ntiqtu i atnfnda I|-x2 *l
funls iyas ma nazaran )'Ithek tertiMen sonsrz kigibndir (o <,1 < l).Mllpyba ehrnetino goF inEqDlylg r.
u
j.J71. itn$l') ei, {rgrg tnls9x\ firnls iyas mr r=o noqbsi etiafnda { sorsuz boy-ok
J* t4firnts iyas r ib mtlqaybt €aelq (bu rrda L < l. < t\ yazatq:
*L ;!=.Y-
(,rft = .Y"€+=.!% ffi
= "
Buradan 4x -.r qiymetiodan
m6lumdur. , -+ +0 olduqda
btifrde olunmr4dur. r+l>t oldugu'2intsqmhlh frnks iyanrn tsrtibi +
x^
tunks iyas mrn artrmmdan boyltkdur. (1. ,.,) '1\ aewt l. t..raolf,mda yrf,rllI. Odur ki m0qayha ohmotins g6n ve rilan inbqal da
YGtlr.
ttts.*i eI xPne x
Halli lnx=t evezbrnes ini apoaq.
x=et +&=e,d: (t=l olduoda l=o{-r=+- omrqaa t=-
oldulunu inbqnlda nazen alaq:.p & ."(r- eIr ---L = l:--4i xp nt,t x '6 tt
301
limr-++()
, + +O olduqda
+=,*),p > I olduqda ,(r-p) firnbiyas! r -+ +- oHuq& j *or"ti f,tnks i.yas m
rl,zaDn sllEtb azalr.(ixtiyari , > I 09Un). Odur ki, p>1, q<t oHuqdaintqnl yrtilr.
tttc"*i e'e xP(larY (hh.xY
Ec& lnhx=t evazbm*hi apoq.hx=e'+x=e"'+&=e"'etd
x = e oldug& 1= 9; 3 = ooHuqda be I =.roOn&
+- -(t_ oyl .!r_q! * _*f,r,t*t= I "o 'lP
'- tr\F'
,J{o olduqda ft)=o'(;); p=r ohraq>r 0e0a
^,=,(i)(f > t); Oger q -ixtiyari heqhie&4 p * t, be, on& p>l obsa
,,r)="r" rr;r.rt =r(+), rr, o
Beblikb, sgor:a)r <l; p=1; q21.
b) r < 1, a-ixtiyari bqhi ede4 p > I oHr4da intqal y{ftr.
]J77.*f e (or.ot.....o,\!*V - "iPt F - ozlP' . -.lt - ",lo'
Eelll x = aya2,....an n0qb b ri otio fudar)^n=r]=
*l tr=tr,..,,1.\P-atl- )(l
.r-+o oHuq& it )=o'lrr,*r,.-_, J
30t
Pi.\ fPi>L (i=12,...,D)j=l
m[qayiss alametine g0re int€qral ylpln.
lit& *i
," 1' -r1o 60
flc& l!l= xdlt-{' kimi i|are edek. r -r +0 olduqda
rt't=r'[-!,)\t-" )
.r -+ I olduqda
/(,)=o'[--+')U'-{-P l
.r -+ ro oldrqda isa
i6)=,.(;#ir)olar. Ona gOre
- a <ti - i <ti -(a+ B)>tYe y8
a>-1" F>-l; q+F<-lolduqda integal yglr.
ASaNe iteqrallou n&tbq ua y gaai yg masm tdqiq edin:
1379, t=*f tu'&. cdstrris: pin{>sin2.rIr
Ealll Verilen iateqrah
r=i!e'*= j'io'a+f 9'6=1, *1,'^ r 1.rUU
kimi yazaq. a > 0-i:rtiyari qeyd olunmug adoddir. Ittrr fiuksiyasr (o,a]da
kesilmaa mehdud funksiyadu. Odrn kr, Ir -inteqrah vor. Dirixle alametino
g0ro l2-inteqrall yfrlr. odur k! I -inteqrah yrgrt, *i"h"a
iorcqol,6,dntiilr.
pin rl> sin2 x
oHu[undan muqayise o}unetine gore x-m btltun qiymotlorinda I -inteqlahm0tloq da$ln.
309
13a1. +f J; cos x eI r+100
HallL x -+ +a olduqda
, aleg- r+ 100
olur, cosr funlsiyasr mahdud firnksiyadr. Odur ki, Dirixle alametho g0reinteqral yrprlu.
t:11 = J __lllcos;ld
0
inteqralmr todqiq edak. B0t0n x Jer 09lln, l+cos2r
icos.Il>(:os'x=-dofrudur. Onda
r- . +.o -t,r! I 4' drrl J --1{1-"6 216 = 1. a1,' 2ix+t00 2 i.r+100, + r-6 olduqda
fi ="'(L\r+100 l.Jr./Odur ki, t2 -inteqrall dagrl[. 13 -iateqrah iss Dirixle olamatina g0ra yrlrlu.
11 212+13gertina g0re verilan inteqral mlttlaq da$lr.
+4I3EI. t= I,P sinlc b, (c+o)
0
HalH
evozlemesini apareq.
1 l-,a=f 1to=Ltc d, O<r<.
q
Bunlal verilen inteqralda nezere elaq:
I +P sinrdr l1 sintdt I +P sin t/'
= Et I .a- = a!;:a. Et j ;* =', *',
14tq14a > o-ixtiyari qeyd olunmug adeddir. r -+ +0 olduqda
3r0
on& P*l r-tq
ihdi l2-i
inteqrainr tadqiq edekBuna g6re
kimi g0sterek Brnada
Yuxanda qeyd
aralt[nda
r)sinr -.1 I I
-=u
t-t.-p+llP+tl;- c t,-, J
olduqda t1-inteqrah lr$ln. 4I < -1 olduqda isa da$llr.q
tadqiq edak 1- P+l >0 olarsa Dirixle elamatine gOra I2c
ygrlrr. Z! 2 t olarsa, tz dalrlr. Belelikle, le] tl . r ofarqa. I -inteqral'
s lc I
yrf,rlu.' lndi I -inteqralmm mutlaq yr$lmasrnr tadqiq edek. Yani
r=fTFi''qlql i '-ai!ttl
I= ir +Iu
'=r'il-^4' '=i;f#g4 lqetdiyimiz kimi P + I , -l tHenirsa I1 yllrlu. (4.+o)
q
isin xl I-J---)- .
-
;Pll - ttl!rq19
olduEundan l-P+l rl ola$a, yeni, p jl <0 olarsa Iz -inteqreh yE r'-qq
Belatikle. -1.4a!a6 09itn I-inteqrah m0tlaq yrprlr. p va 4q
parametrlarinin
o<P*l.lq
qiymettorinda I -inteqrah garti ytf,tlr.
3ll
21382. lttrl(w x)*
0
Eelll wx = t evedomesini aparaq:
J- =, = -u, = l=r r = arccos I + d = -Loos r t t dr2_t;=o olduqda t=t; x=! oHuqda is. r={€ avezlemesini verilen
2inteqralda nezera alaq:
r=T-+d=i-iH, a.i-p d =tr +tzt tltt'-l tt!( -l a t.lrz -l
burada o > I ixtiyari qeyd olunmug edaddir. l -+.,1+o oldugda
rin, -.1 I I
m="1(,Tlaltnsr. Bu halda Il -inieqrah mittbq yglr. I e (a,ro) oldulu halda
I rio, I I
lf\l- il7-'Bele olduqda l2-inteqrah mlltleq yrgln Belolikle tcdqiq olunm inteqralm0tleqyplr.
net. r=*i*z*Q,\0
Eelli ex = r evezlemesini aparaq:
,=l.t-d=4.r=0 olduqda t =t; x=ta olduqda iss r=co. Bunlan verilen inteqraldanazare alaq:
r = Jl6,rf "*aIr' '
r -+ r.o olduqda f - o of,r. Odrn k! Dirixle elametine g0re inteqral
yr! n.
pou4!n" ="*r,!oL =;tl*1*rr,tt,,, r
oldulund"n , 6164
TE+*,;Ttf.;it+n.atmrr. r > c oHuqda "' ' , ! oHog*a"o
-f +, inEqah dafrlr.
*i *ah't a ineqeh yrlrhr. (l) benbesbliyim gon veribn intqal
'ttmtltbq dslrlrr.
Ihtl. t=*i ,P *-' e, (c > o)I l+r{
Henl Veljba mbqnh fti intsqntn cemi kimi gtlsbtsk:
, _*i xe :ia r * _ir P sin.r * * *f :P sin_r & =tr +tz
i l+xa 6 l+re ; l+xqa > o ixtiyari qeyd oftmmr5 odaddir.
.r -r r{ olduqda
#="[;t-")Aydmdf ki, p+l>-l olduqda I1-intsqnk m0tbq y{ilr. P+l<'loHuqda be \-inEqah daErlu. p.s oHuq& -{-+o ohr' Bu haHa
I2-intsqah mutbq ohadan yr!ilrr. Boblftb, p>-2 olduqda l-intsqtatm0tbq olnadan ytilr.
bhirP xP -.( I )r€ [a{41 !--]- < -::-- = O- |
-
t
' l+xq l+r4 \xq-P ).t-+ ro oHuqda 12 mfitbq y{ lr' ager q-p>l ve ya q>p+1; qsP+loHuqde ba mlltbq da[rtr.
Onda qpmq ki, I inEqralmm mutbq y{doas t U90n
p>-2, q> p+lgertbri 0&nmlilir. p>4 P<q3 p+l oHuqda ta gerti
yE!tu
-- ,h,f, * l)13t5. t= l \ ')&
OrI%&, hcqnh iki inteqnlm cami kimi yazaq:
, ,i,f, * l) r- rt,[r* !]I=i \ r'r** t \ x)e'-l ,i *' i r'
(r)
313
x.+! = t eYazbnqs ini aparaq.r.r2-t+l=0
'='l'l4u t 't l:'- '2_= _t_.lf _4
/\*=l!"L'l;u12 2 ,lt2 -4 )
Aydmdr ki, x ikiqiymetli tryin olunur. Ovezbrpbri ineqrelda n z,rashq:
Axrnnc r ifrdani dOd inbqialrn oomi kimi yazmaq ohr:
| l,-, -,)"., [.!*,]"',r=ll.r'\J,'?-4 J a*'fl,l,2 -4 ) d,
'l i (,-.F -\' L (,*JFu\"Ll.-,,]l.,-l,=r'-'l*7 "'* .
l; J-,_4(,_Jr_4)'-'
- I +P dntat'= ,*t l,
, ^"-r1 simtd'J-' t _/ __\r_t'
2,'lt2 -tlt*^lt2 - ql\./
, -nn-t+f sintdt'4-' LT,r4q3t4
=Ir +Iz +Ig +Il,sittd
- | q sirrd,'=i;tr,17-41ry.
+l2
4 > 2 -ixtiyari qeyd olunmuS ededdir. , -+ 2 + 0 olduqda
sin, :r.f , Ifr.1_-@G=r "LA;l
Belalikle I1 va 13 yrglr. r -+ +.a olduqda
=,(#)
='(;)^Fu(,.Jlu)'-'
2-r,>o olduqda 12 ylErlr; a>0 olduqds l4-htEqrah yrf, r. Belalikla,
0<a<2 olduqda r-in@ralt Ytllr.IJt6 Agatrdakr baraberliklerin do$u olduguou gostorin:
l,+qca) i4=c b) j ^; =q v) Jsinrd=o1, ir l_r.
HaUL
", J,+= rr,l1+.i*] = ;*["*-i . n1,li1 = ]| r,r = o
,T# =
"9,f' !', 3.J"#)ri * =
= i;*,i"t i,'.'liJ*"].i,t*'Hl,,. =
=l 11,n 6P:j].l '* "lt*'l=o2 s-ao lz+El 2 t--+to ll-Bl
o fit,*=g'- f,,,*=ri' [-"*41r]=
= ,l5;["* {r' ] = lS_[cos(-
.r)- cos(;)]= o
A Safuh inteqt al lan taPrnl
= i;sJ"l?l.,,H1- "lt#]. i,vJ"lii#l -'"lH] =
315
BS7.-r ^-L -'o x''3x+2ECIL x2 -3x+2=o tenliyini hell etmekle, kOklerinin a=1, a=2
oldugmu alarq. Demeli, a=\ x=2 noqtelerinde inteqralatI fimksiyasrtra gewilir. t > 2fuiyari qeyd olunmut edad olsun. Onds
&&&&7-x+r=G:S:j= ,1- ,1
Tl* t. J =
"'*l'E -?r' . +=il i. " . "l#i.,,l.
* ,6 "E-4'=-b2+
lin frt*t*rt*'l* ,r ,ol'-'l=t-r- lx-llt s-rol I-a l-p.l a-+ro lA-lllt-+u
= -rn2 = hl2
ns&l el.r ln r,
fla,lll
t_-tlim [n]-l = 0
e-+ro I c I
3r6
i* =iy =
"Y,*[""'i' . "("]i' ]
=
=,t Jr,(r,(r-'))- {r"}). r+"a- r,(u(r+ ")]=
= ,- hlEg:eX=c-+ro lln(l + el
I3se.tr ):+eHclll Txife gbn
.IIi * = tr-i,n " = x{1,#. i,#)== E *l-" *t!,. t'l'. *{:, ]
=
= ra f-" el - * "
te ( t\ + f,ul +,e'z 1- l"l' <-'f l)= t. T ='
l3eo.'i-",sre
fl.lll+. V"qg =u'
J-ase =l e, =a,.ll+r'
= ^f*"*t!, i,n")== Pf'-* +^oc^('i' lni."11, ]
=
=-l n, hE4=oz t-+o lt+ lzl
qrtrolar:
Qcyr*t r;xlnst brlqralLt hazouY---%
dv=&ll_,=, I
Bn. T4 (a>o).OII
1392. Ilnr/r.0+?&I ----;
--.o I + x'\e-l {l-r'+@&)-a;;;f
cavab: !.a
Cavab: -1.
Cavab: a.
C*ab: r.
cmab:?nz.3
1393.
1391.
1i95.
3t7
1i96
1397.
139t
I0
I0
I0
Cavab: -o3--.u'+b'
Cavab: -j .a:+b"
&l-7e-e ccsbx& (a>0),
e-6 sntx& (a > O).
4ttuavaD: ....-.
3./3
$5. SAHOLORIN HESABLATYMAST
1.Salonin d zbucaqh koordinat sisteminda hesablanmastKesilmez l' = _vr(y) ve r, = _v:(r) (J,: (r ) >.r = a, .\ = b (a < b) dW xotleri ile ehab olunmugArA2BtBz (Sekil 1) sahasi
1't(x)) vam0stovi
(r)
d0strru ils hesablanr.2. Parametrik Sakilda verilmiS ayrilarla ahata ohmmq mtutavi liqurun
sahasi.C qryah ayrisinin tenliyi r = x(t).y = I (r) paraoretrik SekiHa
Y€rilibse, onds sahe(9ekil 2)
(2)
dtlsturu ile hesablanr.
s = {"o
lt,ri - y1e)l dr
s =.['r(t)r'(t) 6 = [o'
,1t1y'g1at
3tt
3. Potym koordinal sistemindo sahanin hesablanmast
o: it l eyrbive iki yanmd0zxetbrb to = a,q = p(a < p) alnt>
ohmmls m(bt vi fiqurun sahes i $ekil 3)
s =i.|! o'ki a'o (3)
dusturu ib hesabbntr'
1399. y = x2 paobohs r, x = -1, x = ?dnz xetbi va abb oxu ib
ehab olunmrS frurun sahee ini taPn'---XeA nitantan fqurun sahi (l) d{b nrnr ib hesablamr' Bunda
)r(x) = 0 oHu$undant2 i-:
5=l g,,,tr) -yr(x)l ri.r = | x'zdr=3- J-t*'- )-r
tskil3.
parabohdr. Birincininoxu be oy oxudur. Bu
II(N, s=y2, .ry=x2
Ileltt AYdm g0flh0ur ki, her fti aYri
simrretriya oxu or -oxu, ikincbin s immeniya
mEoEfin bidnci d(bturundsn btigde edek:b
s=I[yrG)-nG)ts
r-in daybms inErvalrnl tapaq:
,r=*, ,r=i-\-..=o=-lt-l)=o= x=0, x=a, ol,3q'
319
tskil4.
'={- *b=l* r r!=ir-*=*110L y=y2, aay=2teW Panbob ib dlb xetin kes imes inde ahiin koodinanthrmr
Epaq:
x2 =2- r + x2 + x -2 =O, 4 =1, xZ = -2, -2< x SlMetum dtbturdan istifida edek.
'= j!-.-*h=[,. +-+)_, =(,-i-r-
-l.-n-r*!)=r- 5 -6 -8-5+16 -r-z= I\ 3,/ 6 3 6 2 2
1102.y=2x-x2 aythiva x+y = O ttlalefi ib ahaa ohurmr.qfqunrn saha ini bpn.
Eefr. Owaoe ayi ib d{tz xefiin kes irc nOqrebrini iapaq.( Y = 2x - x2'['+Y=6'
s btmini hsll e&k.( u=-, lxz'3x=o
+l ' . ^ .+lx-o.x=gl-x=2x-xz lr=o.r=_,
(0,0) r,a (3. -3) tl{lz xetb eyrinin kesirn n0qtsbridir. Onda axunhnsahe
,= /'rr,- r: + r)dt=lo'iar-r.)ar=(+ f)1, =+-+:+=i1103. y = 2'eyrbivo y =2, x = 0 d0z xetbri ila ahab ohmmq
fqunrn sahosini tapm.Ib& Owece ayri ih d0z xetin kesime n0qbsiini bpaq:
y=2,,y=2+2*=Z)x=!0 <.r < 1. Odurki, axhnlansahe
, : ['<, - 2.)crx = (r, - #)1,"=, - #. *= r - #11O1. y = 4 y = a asin2 r, (O<r<z).^ArrH, B irtupi dBtudan btiQda edek
s= j(,*,io2,-,! ='1,n, ,a, =00
= lir, -"* r,1, = 1l.,- lrio z,)' =26' ' 2\ ' lo
= lf" - l"h2'.) = "2\ 2 ) 2
-t110s. t=-i. 1, r=o
IIatE x-+q olduqda y-+0 olur. Bu adi qaydada kvadratlman
fimksiya deyit Funksiya cllt funksiya oldulundanAA
sr= ly\,\B =zly{'\fr-'l 0
s, = zotl -!- = zo, *"re 4' = 2a2 qcte !iio' + x' ab o
s= lin s,r =zo2!=*2/-+o ^ 2-: .-:
1106. 7 +; = 1 elipsi ile ehae ohmmug fiqurun sahasini @m.
E ttl , = ! t;r- ,; kimi tayin olundugrndm axtaflIan sahe
agatrdak kimi olar:fob --=-"--=- - I x = asint I.S=4f -.la: -az fla=t ^ - r=" -Jo n'- UY = a66s1l1l
+ari
-
=-l va, -ozstn2to -to
1107. y2 = x2la2 - x2).
fldIL *b2 - t'z) trdlai l{ < c oldr4da miisbet firnksiyadr.
y, =t@ -,,\ ('=*,[7-,r)firnksiyasr qryah funtsiyadtr.
321
acostdt = +a [i ,or' tat =
ti=zabl 0 + cosLt)dt =
.r0
rrin= Zab(t +
isin}t)lo = Zab'1= nab.
Bu koordinant oxlatma nezsran simmetrik fiuksiya oldu$undan
Derneli S = 112.'l110& y2 =2p1, 27 PY2 =*c- PY
Hellt y2 = 2pr funksiyasr .x > 0 qiymetlarindo t yin olunmus
finrksiyadr. 27 pv2 =sk- Pf funksiyasr ise : > p qiymatlerinde teyin
olunub. Oyrilarin kasitme nOqtelerinin absisini tapaq lkinci tanlikden
z rlz!- p\13v =iL-;)taparaq, bunu birinci tanlikda yerine ya.,zsaq alariq:
l2('- d1' =2,,,L s ) -''Burda
*-P) =,3
evazlamesini aparsaq, z-e nozaren kub tenlik alanq:
,3 =!p2"+2prva ya
"(, ' P\' + P)= zPz(z ' P\, z=-P
kOkt bizi temin etmir.
z2-Pz-2P2=gtenliyini hell edarek, z1=2p vQ z2=-p (yararnr) alanq' zden ewelkideyigan .r-a kegsak, x=4p alanq'
,=/Er,-,ri- \27 P'
Funksiyasr x>p tlgtto tayin olunub, pcr<4p aralt$nda
"['*.'EU- 'l'127 p'Cisim ox oxuna nazeran simmetrik oldulundan, alanq ki,
322
= lo,'!"in, rnz d = -to2i*"2 ta(*"1= -**t' tl' = -T00-10
, =,{i,r.+ * i fla.i - 6o -,,, ]-i
=
='{'u;i''1, .l*i'' Gt ?t 'tn:}=
=,li i, -t," r)=,,',;(i ;) = Y1109. At2 +28ry+cr2 =1, (aro, eC-a2 ro).Halll Yenlantanliyi r-a nezaren hell ederek, alanq:
, t-lec-e'l'>oOHuqda -r =,r(y) firnksiyasr mttsbat qiymetler alr. Buradan
pa=.Q'\tc-n2b
alanq. Saheni fitr!\- rr0)1b dOsturu ile hesablayaq. Burada
Bir qedar sadelegdirsek:
s = 2 i 174-;,T d = $_ !O=,, =
ir = a.inr, JF= =,[,t -n "A=6csr, r = -Dotduqdar =-4]=l-r=
pt=rc*,a, x-b olduqdar = I I
I=),QE - a, tl., * =!.Ee= =#ttro. ,'={, (asissoicl), r=2a.
flalll Bt ox -oxuna nazer l eyri simmetrft ayridir.
n@sinde gyrinin maxsusiyyati var. Ona gore ayrinin sahasiqeyri-maxsusi inteqralls ifrda olunur:
3
s=27 ,'e,,o (u-r)i
x =2asir2 t avazlemsini paraq:&, = 4osiol @|td:=0 olduqdar=O
x =zaotduqd,r r =I.Buahn inteqralda nazara alaq:
t1
t = ril4 *,,i^ t w a = t u'l "io'
a ='o.J2awt o
!a= +o'|Q- waf a = dl( z*a * *' t )a =
ikinci ndv
0
= +"' 1(|- z * z * i* "\f = *' (it - sn a + lsin *)l = u2 t,rrt., = ono* W - E:V,y=o (trattsis).
vEdll Agkrrdtr \L o<y<a qiymetlerindo r srfirdan sonsuduta
artdrqcq y azalrr. Verilan ifadmi diferensiallayaq:
*=l , #-?Fa.,L=l'.la
-t t .1"-r )'
It.--t..L=
,,li-f l'
ffiy
-",1? -f -? +-7;W--4',."!e-v]|-fl€-
ot -t;;W-l.=l;;14
3U
=( -,' *---L--E-v ^\r'lo'-,' .'1" -,' )' Y
Aydm gor{nlh ki, r-m m0sbet artrmr, y-in mmfi arlDma uytundur.Odur ki,
,=.iw=iW s="1,@-'a-oo!ob=asml x=0, r=0 : ,:I ' l=]i*'?a=\ig+cu,ux.=-lat="*,a, x=o r=il - 6-" - 2'ov' ----''
,=o'1,*1*rli -?n.2\ 2 4o 4
1112 y2 =, "-o, (r>Q z>-2)b* ,'*' f
flcA W Ox oxutra nez en simmetrikdh, r-+ro olduqda l/-+0olur ve fiqunm sahesi qeyri-mexsusi in@ralla ifade olunur:
7+6 -2s=2r #;,
1113, y=e-'pia4 y=0, (r>0)flaill
*. (t+t!s=, I "-,W#k=0 Lrtrurada, :-tz=r avazlamesini aprsaq, onda
s ='i e-bie-, sintd .f-0 0
Burada hiss+hisse inleqrallma drbturunu fti dafe tetbiq asst alanq:
s=f "-*'L!gl.o*{ 1+e-* S"-* -r-=021*2ia
4tt2*n+2 2 n+2
325
_l+e-t I _2 l-e-o
1111. y2=2r parabolasr
nisbetda bollk?EaIII Dabanin sohasini s1 ila,
sohesini s2 ila i;ara edek. Aydmdtr ki,st =tz
Parabolanm gevrc ilekasigmesinin absisininkoordinantlarm tapaq. BunegOre
? +b+t =9bnliyini hell edek. Bu tanliyinbir kdk[ var.
(x+tf =t x =zlndi asanhqla S2 -ni tapanq:
sahesini hansl
parabolik ve dairevi seqmentin
,t( r .r\e 2le2 +e zlt.t---;7 , ;\2r-r1", _"-,
It.i
| -1,22
x2+y2<8
,, = {* -'t1^ct - *b)= I,. " i.ili,
i.r;+ -)=
= {,..!,a -ia= *7= \,"* i
- i"tn -,, *)2-lJo - ,' a -irt ar"trnr hesablayaq. Buna gdra t = 2^,l2sint avazlamesini
0
apamq.1=e r=0,
-- ', ,-E
& = 2Ji ccl&Bunlan inteqralda nazare alaq:
326
L,_-ai"fi7 t ='1,8- t
"i,?'' z lz,*,a, =
00trl4^4
= lzJioosz o^fza = t ft +costuld =00
= ol,*l,rrli = I L *!\ =, *z't' 2 )o \1 2)
Axnncr ifadeni s2 -nin hesablanmasmda nezare alaq:
s, = z{2, * \ -, - r)= \, . l) =io, . rlt,-2,-!\lz,*!\=
(s, -s,):s, =[E; 3/ \ 3)
=(r*-!\.1r,*!)\ 3r\ 3)
Poametrik sakilda verilmis asafidoh ayrilarla almta olunmuq
1i qurlar m s ahes ini I aPrn.
1115. x=a(t-sint), y=a(1 -cast)(0 < t <zit) sikloidi ve
/:0 absis oxunun hissesi ila ahao olunmug frqurun sahssini tapn (pkil a).
floIlL O ve A noqbleri paramebin uygun olsraq ao = otte tA = 2T
noqtoterine uytundur. OduI ki, axtanlatr saha (3) d0sturu ila tap r.' flr t)r, = I o (t - rost)n(t - sint)'rit = a'.1 (1- cost)zdt =
= n, f"t, - zcosr + t'oszt)dt = o' .[, (r - zcost -\i9)* =
= * I"'" (1- zcost +)coszt)at -,'()t -z"i,r + f si,,zr)l'" :=rr.1.2n=tnn2.
2111d r = 2t -r2, y=2tz -t3flCfr. Bt ayi koordinmt baglan$cmda 620{dtnU kesir. (r = 0, r = O
olduqda r=2, y=0 olduqda). Sahani hesablamaq 09Un
"=i'd*>'o-rt>'tftd0stmmdm igti&da edak:
' =iih' - t\u - t z1-@z -,t\z - ufo =
=!f$rz -rrt -*t +3r1 -4t2 +4t3 +2t1 -{h=26'
= ;i(,' - *' . "'h = :(f, - n . :u):,,= :(? -,.. l, = i
107. x= d(2555t-1p6a\ y = lzsir:. -snu)flolfr. Cisim q"palr ({o)=.r(2t\y{0)=r(2t)) xatlari ila ahato
ohnmugdrn. Odur ki sahe
' =1il*>'o-,r>'& (,)
dllsturu ile hesablanr.
r' Ql= lzrxrsr - 2cau\y'(t)=d-zsnt+zsna\
Bunlan (llde yerina yazaq:
s = j'fi.{z "*, - "- 2t\42 w t - 2 w a) - f- z sin t + 2 sin 2t)lz siir t - su u\fu =
-- i'i Q *', - z *jll cos, - oos2jcc, + cos2 ?r + 2sin2 r - ein z sinr -
32t
- 2sinu,s,,tr + sl,2 2h = o't1z -l'*lrcet + 1- 3sin ?Jsinrp =0
= *2i1r - *4a = # Q - "i"tff = aiHr& r=L,g.! t, ,=lr;^t r, (r' = o' - a')trurnuin evolwtasr).
Ezlll
t=-'lx>'<uJl*l;d\4 (r)00
d0gtunmdm istifide edak:2t 2* ^2 ' '2 ,2 t*rd:t-\ ,*o ,"5' ,a =t= lrr4= [;*','17.i,-
-- -- ab i)
=frT *',,;^' xa =*T,xy * ^ =
= z"n .'i6 - ** * w)t - ,,su *ai)d =l6a, 6'
= *'i7, - * ". *u - f,*a' - l*aP =
a,
=#(, - i,"*. l"az, - l,-a -1"""f, = * * ='#osim2 t
1119. x=owt, Y = =----;-.2 + 8ln,
Ifc t t -nin o4ll.r-ye azelmasr ila I aJm -a-ya qadx azaln, v(j
azalmr. r-nin 0daa f -Va daVbmasi ila v Gdaa i-a *'
x=a#t, "=!"t',2+S,nt
tenliklori (a,o) ve (- a,o) nthtclorino artan qapalr ayrini tsvir edir' Sgheni
'= !*dtlsurru ila hesablaYaq:
379
s = ? '"h" (- o.i n,1* =-o"7 " '' 4 =[ 2+sint' ' L2+siat
=-o"i("n'r-z"ir,*l- t ')a= k"f I a-g^'=6\ 2+sint) 6 2+sin,
= *,li futl-; r * ; tri) ;1-, *, =
ll"r* ,) * n l't z* z) * q)
=*l*0p,1**np,|1*=
=t€ ( o - o *' * L\-c*, =Y4 -e*, = -r( )1 - s\J3\z 6 6 2) Jj \J3 )
TanliHari polyar koordinatlarla vefilmg ayribrla ahata olunuSfiqurun sahasini tqm:
1120. p2 = a2coszg (Lemniskat ayrisi).Halll Btt ptos.P = Ove psirup = 0 d1z xetlarina nezaran simnehik
qapah eyridir. Odur ki, axtanlan sahe
^ -.1 ..r !J t rr a'rt a2 lt a2
i= i.lo o'av -- 7.lo,otzs,1* :
n t'",*lo= n ., = o'
taprlr.1121. p = a(1 + cosg) (Kardioida ayrisi).Eelll Oyri psinq = 0 d0z xefledna gdra simmetrik oldu[undan,
axtrnlao sahottt lx
5= | a:(f +tose)? d,O = a: | (t + Zcosrp + coszq)dtgt =.Io .lo
' o' [""
{t + zcose +!::fY) a,e = ,, [" Pr+ zcoso +)cosze)d,p =
3 1 t' 3ra2= a'(Zq 'L zsiruo +
OsinzV)lo = , .
7122, r = asn3q (u9 yapraq).Helll Uglagekli gJtl0n 09da bir hissesinin sahasini hesablayaq:
tat
1 = r;1 t-",{ tt r, = li 0 - *" u rv, = !(, - l"n u r\: = t + = *,o-*2
u23. r =:4- Orabola), o=i, ,=il-co89
.EdIl Sahoni
,=11,'@v,
dtlsturu ilo hesablayaq.tt
,=Pi-11 ","=4'lr'*,r="-z)p-*"ef zi(z"are.}.4 4\ 2)
,u
= (\,. *r, z)4* t) = *(* f . i*' Z\1,=
= ({a. !l{a.,Y -,1} = (Aa. z li + a + tri + t - r)=
=Sl,a.u)=(loa.,\1121., -_ -__!-, @ < a < r) (ellips).
l+acosPIIaIII Satreni malum d{tshrru totbiq etmekle hesablayaq:
'=o"i o' ^=o'1 da -'"-I l1r+a*'ef - l(r*e"*ef
M0ayym gevirmalar aParaq:
t * ", *" q = *"2 9 *
"in2 | * "(*"2 I -rir,'z f ) = 0 *, )*, z L * ( - "\"in2 t'
(t+"*"ef =[O-"t*"'f -tr -,hr,'t).Axnncr ifadeni saha d0sturunda nszers alaq:
i
331
s = e,i -------!e.--------i = rr,i !. d N4of(r*")"o.'f *0- "1"""r)'f(r+")*(r-")s'f)
=lt='',==o)' :: -l= #Titi" =I,["i='l=
= -[+{ ru##-n * = ft '*,olt' . ffi. i i* " =
no2 g'" 4p2e *i & p2(t+ s\ 4pza ..=
t ;F. G;F- T -,4 I [;;I=E;F- (;F^-(rtr.4.;-*\:,"=t# #=#
7125. r =3+2qs9.flalli Bu fiqur rsinp=0 doz xerttine nezeren simmetrikdir. Odut ki,
t ="11, * r* rf o, =1(e * n * o * t *2 o)e =oo'=r1 (t + t2 oos s + 2 + z or.zg)a 9 =o! (t t + 12 w 9 + 2 w fup 9 =
00= (t g + a sir, s + sA zalfi = t t:t
It2d r =L,, =.]-fo.r=il.p sln9\ 2J
.Ealll Melumdur ki,
sinpce (r.r.;).
*i (,'(";)Oods
oldutundm
t2{ r l). rlz t'= , !l"a;-a f'=do*i
332
,1Orl
Buradar ( l\ 1 -- t'cose l),.'I*l'o' - ; )= r" Trd'r'" - ;J =
1 .. ecosa-sina I=-: tial -----: =1 lim2e -++4 tsm6 2 e -++4
ag2 3 slrnm4dr.
Qahstulo:
Flqwut szhasl1127' Tenlikleri y2=2x+l ve x-y-7=o olan xetlarlo
fiqunm sahesini hesabloYn.
,-*-,-'1..o'("s)
"2 *o'(ra)
=0
meMud olan
cavao: E.3
112& y=-x2 +4t-3 prabolasr ve (0,-3); (3,0) noqtebrinda ona gskilmig
toxunanlff srasmda q"lan fiqurun sahesini hesablayn'
cavo: |.1129. y2 --2p, par'abolasr ve absis oxuna 1350 -li b,ucaq emale getirmekla
iamin puabotaya gskihig normal ih mehdud olan fiqunm sahasini
hesablaYn.
ca,*,!i.1130' y=r2 ve /=J; parabolslan ile mshdud olan fiqurun sahesini
hesablayn.
cavo: |.1131. y2 +tr=16 ve y2 -24r= 4E prabolalan ila mehdud olm fiqutun
sahesini hesablaYm.
1132. n x3! = xz va y = ? Praboldfft ile mohdud
ld1"..
1a_c*ab:'-3J6.
3
olao fiqunm sahasini
Cavab: 21.t133. -2
4
r=!- nraUofes, *2 *y2:8 gevresini iki hissaye bolm gdllr.
Qevranin her iki hissasinin sahesini taprn.
Cavab: 2r +! ". oo -!.
11J1' y2=6.r parabolr"r, ,2+y2=16 c€vresini boldukda alnanfiqurlmn sahelerini taprn.
a _cav*: l$n + J1) va !G, - *r.
1135' "2 -2y2= t' or**^ , ,2 * y2= a2 gelrosini ug hissaya
b0lnttgdtlr. Bu hisselerin sahalerini taprn.
"**,,,1I -+r,rr. Jrr]; .,lI-S^t,. a,),"
,rl4*9ntS..rzt1.L3 4'- '',|
I$6. ..2 2
'o * ,' =t ellipsi ila T- ,' =t hiperbolasrnm kasigmasindan
aknm eyrixstli fiqurtarm sahelerini taprn.
Cavab:,S1 = q =, - !nl- Z rcsio rEl,
O,+6, 52 = Z17e - q;.1137, 1 ,2
y = I + t
xatti ila y = I- parabolast arasrnda qalan fiqurun salresini
hesablaym.
113& y = 4x -l)2 xa6i va absis oxu ilehesablayn.
ca"ab: l-1.23mahdud olan fiqurun sahosini
Cavab: a.12
mehdud olan tiqurun sahasini14:19. ordind oxu y3 5 = 121y - Iy xatti ilo
1111. (y-rf =rt xatinin iki budalr va r=+fi qunrn sahasini hesablaYrn.
hesablayrn.
Cavab: 1.t2
1110, Absis oxu ve y=r-r2J; xetti ile mahdud olan eyrixetli trapesiyanm
sahesini hesablaYn
Cavab: I.t4
d{lz xatti ila mahdud olan
Cavab: zll.7
1112' 0-r-2)2 =9: xatti va koordinat oxlan ila mehdud olan fiqurun
sahesini hesablaYn,Cavab: I (fqur, sahel*i berabar olan iki hisseden ibaretdir).
11$' y2 = x(x - l)z xetti ilgeyiain sahssini taprn.
cavab: 1.l5
ll1tt, yZ = (1- ,2\3 qapah xetti ile mahdud otm fiqurun sahasini tapm.
Cwab:Ttt.4
H15' y2 = *2 -14 qapak xatti ile meMud olan fiqurun sahssini tryrn.
cavo: f.J1116 *4 -d +o2r2=o qapah xatti ila mahdud olan fiqurun sahasini tapm'
gorug,4.t
1117' y=ex, y=e-x xetlari va:=l diiz xetti ile mehdud olan fiqurun
sahasini hesablayn.
cavab: e+l-2.e
111E y=1t2 +2r)e-r xetti va absis oxu ile mahdud olan eyrixetli
tr&p€siyatrn s€hrsini hesablaym. Cavab:4.
1119' y=2x2e' ve v=-xler xatleri ila mehdud olan fiqurun sonlu
335
hissesinin sahesini @n
115L y=br va y=52,hesablayo.
usl' y=bJ ve y=xtnx
hesablaym.
1152' odind oxu, y=t$ ve v =!"*,'3llgbrrco$n sahasini hesdlaYn.
^ . 3-21a2-2] 2 2cavaD: :--16
xetleri ile nehdud olan ayrixatli
1153. Absis oxu vo ,=arcsinr,/=arccocr xetleri ile
sahesini hesablaYm.
O"aO: ]*lrf .
mehdud olsn fiqurun
Cavab: Jf - t.U51, .r=a(t-sinr), ), = 4(l -cotr) tsikloidinin bir ta$ va abois oxu ilo
mchdud olm fiqrrun sahosini taPm.
C*ab: 3tr a2 .
1155. r=acos3r, .x=asin3/ astsoidi ilo mehdud ohn fiqunm sahasini
tBPm.
cavab: lr o2.t1156 x =2acrst - aow2t , y =2asilnt - asrmZ kardioidi ila maMud
olan fiqunrn sah*ini taPn.
Cwsb: 6* a2.
cavat: f6.
Cavab: !.l5
1tts9. p = asfr4 xefti ilo mehdr.ld olan fiqwun sahasini tapn'
cavau: {-ze'
xatlsri ile mahdud olan fiqunm sah*ini
Cavab: 3 - e.
xetleri ila mehdud olm fiqunrn sahesini
1157. , =3t2 ,y =3t - t2 xetin ilgeyinin sahesini @rn.
tlSL t=? -\y=F -r xefiin ilgayinin sghosini tapm.
H&.
. 1161.
1162.
1163.
t1u.
1165.
1166
1t
p=dCe (a>0) xetti ve 9=Zsahesini trytn.
p = 3 + c,os49 Ye P = 2 -cns49o@ hisssinin sahsini taPm.
c* *, ((d6rdbfekli sor).
p = aocssq xotti ile mahdud olan fiqunm sahaini @n
cav*: ff.p = 2a(2 +qsq) Paskal ilbizi ila mahdud olm fiqunrn sahesini tapu'
Conb: ltz. a2
duz xetti ila mehdud olm fiqurun
2
cw*:2@-r).txetleri ilc mehdud olm fiqrnlann
cl^uabr 37 x
-s.13 .6
p=2+cos24 xetti ile mehdud olan fiqurun p=2+sinP xatti
xaricinde yerlegan hissasinin sahasid t!pm'
c""rb,5lf.l6
pz =a2 rrr'rg x*ti i.b (r'tam ntlsbet adddir) mehdud olm fiqunm
sahasiai toPm.Cwab: a2.
o = orinl I xettinin xrici va daxili hisselari arasrnda qalan fiqurunJ
sahesini @m.
c**, ? 54, . Xati qurmaq 0c{n p+in oda 3' 'ye qedar
deyigmesi nezeradm kogirihelidir'1167. p=.[l-- ? ,s=oea, *Jil7 xdti ila mehdud olan fiqurun sahesini
' hesablaYm' cavab: f,.
: Poltot kootdt to $a k grtowo frqu'brl" sahoslttl h*oblayu'-116&
@2 * y2)z = o2(rz - y2) Bernulti lemniskatr ile mehdud olan
fiqunm sahesini tapn
1169' 1t2 + y212 -o2x2 -bzy2 =o xefti ils mehdud olanpoderasf ) sahssin i taprn.
g^r5t L1o2 * b2)
1170' @2 ty2)3 = 4a2xy(r2 -y21 xatt ilo mahdud olan fiqurun sahesinitapm.
Cavab: a21171. ,4 * y4 = *2 * ,2 xetti ile mohdud olan fiqurun sahesini hesablayrn.
Cavab: ti1
$6. eovstrN uzuNLUGu10. Qdvstn uzunlugu duzbucaqh koordfure sisteminde Hamar
kesilmez y = y(r) (a < .x < b) eyrisi ile verilrnig qdvs{ln uzuntulu
,= r! f,a,,1,1a,dosnrru ilo hesablanr,
2. Oyrinin uzunlu lt parametrik gakilds verildikde.Oyrinin tanliyi x = x(r), y = y(t) (to<t<I)tonliklari ile
verildikde q0vsnn uzualulu
,= [,'p1,t*,,1oa, (2)
d0shrru ilo &prlr.3. Q0vs0n uarnlu$u polyar loordinatlarla verildikda.Oger p -- p(p) [a S g < f ) kimi verildikde q0vsth uantrlugu
atalrdakr kimi trprh.t8
-
L = .1,
,t o'zle) + p2 (d dq
Csvab: a2
fiqurun ("ellipr
(1)
(3)
1172. r=acosp,,=fcosq+sna\ [r(;r).r)
Holll gakilda g0starildiyi kimi mesalonin gertinden aydmdr ki,S =Sr +S2
burada
,, = f;;g*"' u, = 4r1y, * *"rrY, = +1,. i"^ li = +,, = | i 1"^, * *, rY o, = "]
ol
0 #in2oYe =
-I 1
= *1, _ i**)i,= *( i. l)= * *
1
- -' . ra2 o2 .# o: -@-th2S=Sr+Sz= t* S- 4= 4-+ 4
!173. o = rqctgr eyisi va iki ,=O' ,=?l90alm ila ehate olunmug
sektorun sahosini t4m.flclU
9=0 okhrqda r=0'z okluqda r= J3e=f,=3
olar. Melum hesablmra d0sturutrdan istifade edak:
,=1"{rYr'Ot
Burada dayigeni evez edek:
s = l"l,t o' \,\+ = lf,,'l-",n. j7p =
= lf, r *,n*. tf !- * = 1t', .',,
339
J3 b = o.tp av = 12 dlt7 = t 2*"tgr*=l . d. ., l=o Y=i7 v=T
I
=**'t {!**=*I-:'J, t Ji,
,,= Ii7"= !1,_#P==1.4-tnl,*,,11" =1_15a = l_62\2 2 t ',lo 2 2-. 2 --
,,={-1.1",Axnncl ifadeleri
s =j{r, *rr)d{lshrrunda nazere alaq :
' = :(+ - ;. :",. ; - ",) = l( {., -i",)1171. 12 + 92 = 1 ayrisi ila ahato ohrnmug fiqurun sahcsini hesablayrn.EaIII Verilen bnlikdm aydrndr ki,
odllr ki, axhdar sahe wst'
'=i'!'sb'=it'-ia), =?
1175. p=sinlr, 0(r<l e)rrisinin legayi ila ehsto ohlnmug fiqurunsahesini tapm.
Hetll r oa- |-a
qeder rtdqca, 9 o4an t-e qadar rtr. r l-
am j-e ooaer azaldrqcs, p l -dan 0-e qeder azaln. ** *, i\rro,ftinfeqalmn qiymeti a:rErlan sahani verh. yuxanda deyilenleri nszarealsaq, alanq:
3,O
=-1r*uJ'*l'l*"-*=lr lo r'o lt7176 q=lr-rt, P=0 xetlari ila ehaa olunmuq fiqurun sahasini
trym.flettl Bu naselanin hsllinda brmdan awalki maselade oldutu kimi
m0hakime ryarmaq lazmdr. Odur ki,
s = -)?0,'b - *') * = -li,(*' ',t) * = -:(i, iu):,=
_ 1( +f -tz.t\=-,11-1')=-,u.
s -s -64.-2[3 5)--\3sl ls ls
1177, q=r-sinr, g=n x*leri ila ehata olunmug fiqurun sahesini
t8pm.elll o<r <n oldulundan
s =L'1r'11-*urY =l' =u' dY = (l -cosrpl-2d W=dt v=r-sinr I
= }f.'t -.'4' - i1* - "-,w)= *- j'{. -.-'ln =
=v:' """:i
".:l= +- i[+
.' -, l. - i(+.
*.')*] =
=+l+ .l+.,,"'),]=".+
tlTt- , = Z-, q = 3 qrprh eyrisi ila ahata olunmug fiqurunl+t' l+,
sahesini tapm.fldi ,ro olduqda r>0 olar; l=o olduqda r=0 vs t-ro olduqda
r -+0 oftr. Odur ki,
. ldtde=
$+7oldutundan
'=;i,ffi=-'"fffialmar. Osboqradski ltzulu ile hesablayaraq alanq:
' ='-'l' ffit i*'.1) = ^'()- ?) = *'(' - 1)
Yerilan ayrilarla ahata olwmug mflstavi fiqurlann salosini polyarloordinantlara kegmaHa hewblcytn :
1179. x3 + y3 =:qry 1prek61 yarya5).EeW
x=r@39, y = rsingg6t0rmekh Dekart yarpggnm tanliyini aga&dak kimi yaanaq olar:
,3(d 9 +"a1 9)= ulcospsins.
,= *?ri"{ . (o=eri\GOt- p+srn-9 \ L,t
Bu ifadeni sahe dllsturunda nazere alsaq, alanq:,t
'=i!'"=*'1dffi*=+iffi=y2 7 fi+ts3sl 3a2
- - t, $-2a
,iu2
.,o-z ,ot*ricy
z
11E0. xa +ya =o'b'*y')Ealll x=rqsg, y=rsing polyr koordinantlarda fiqunr ohste ed.D
eyri tenliyi
-z- o2' -;;;;R;kimi olar. Odur ki, sahmin hesablanmasr d0stunrna g0ere alanq:
3A
=4.,=,.fu'.uU. Q2 + y2l = 2a2ry (lemniskar).
HeAa= tcjsg, Y= rstnq
polyar koordinant sistemine kegdikda, oyrinin tanliyi12 = a2 sit2.g
kimi al&q. I*mniskat ayrisirsroP=rcosP
d0z xatti ile koordinant baglanlrcrnn nezaren siometik eyridir'
7f4l
s = ?A2 [,2 ds = 2] t".z4e = - o2 c*fi = o2
00
Oyrinin anliHarini patametrih Saklo gtinnkfiquun salosini tapm:
???11E2. x1 +y3 =ar (astroid).
Holllx = acos3 r, y=asin3r, (ost<at)
kimi gdfifrek Astoids tanliyitritr koordinant oxlarma nazoreo simmetrik
oldu[unu nazara alaraq:
s = 1l"Gt-t t\fr
dtlshnmdan istifada edak:
143
, = ^,|a# "t;= o
ih
-,i*;,' =Y =''' =
| = r',=,' :
= -z' f !:l a = u,. f ' ; ! i.
=.,. f &,=
# **,H,,
t = )T b*J r.r""in2 rcos, +3ace2rsinr.osin3 r! =
={T06"0,"^',**"'t"no,fo =tTo"n'r*",,a,={To"t^rzra=
={Toa - *-*t*= *t, +.'")l =* * =*11t3.y=1p (o<-r < 4).
EaaL y' = Cl' = lri. Bu ifadeni (l) dtrshrunda nazare alaq:
r = /'.,,i-ffi* = /' jI;,*=; [('*i')i'('.;'=
+ "#l' = f (,oi- r) = f{ro,m- r)
11M. y =s' (O<, < A)flrru (l) d0shflndan istifrde etsnoklo
t= f Jr+eL *= l.h+e-be,&=00 | .ll*"*. *=&')l-i..E;vrk)=f .-L& r=0 lAt=_--:. y=e- |I ,h*"-b I
= "' ..1.-"1: .1"# =
"o . [*fi - Ji *r,
,, =1,# = -1,,#= - r,i,* . {r;1" -
= -rl,* ..f, ;.,"l + r,,lr + JilAxrrncr ifadani yuxanda nazara alaq:
t = "o ..8 *F - Ji- t** ..ft*-3 * r,lr * Jcl.
ues. ,=lf -lr"v (t<v<")
fletL lfiqral dayiSani olaraq y-i gdtllrak Bu halda
dllsturundan istifada edak
;(r)=;_+.Bu noticoni /-iD d0surnmda nezare alaq:
,=i@I;.=iF-{:.,J-r"=ill;.+)'.=
=i( r- - L\^ =( L*lurl" = 4*1- ! =4.:=i\r-uf-\+'2""'/,, 4 2 4 4 4
I1t6'Y=nfrr$ (0 <x< b <a)'flolll
t72 - x2 2a1 x Zaxy, : n.__iz .G;_lF=
a, - x,Bu ifadeni (l) d0stunrnda nezare alaq:
,t i--ii*z rb a: - x2r=l ir*ffidr=J,fidrJo{
: [,(_,_4=_)*== (-, * rx' )r,,lfiDl" = -, - "tlffil = n1n!)! - t
11E7.y = Incosx (0 < r. n.|).Helll
Y' : (lncosx )' = -tgx '
Bu ifedeni (l) d0sturunda nezera alaq: ,ir
a-t= l^h*!9ldr
rtu& ,=o^o*Jlj -g -V (o<a<ys,)HoA llr'raqral deigani kimi (l) d0sturunda y-i g6et{trek.
-, ,,lo' - y'.t _ __ y_,
use. y, =*; (r=,=?)Halll Bwda,yaa (l) <iUsnruadm istifada edek. Odur ki,
3
,= *i ,=,lp,-s),Q"-rh
; 1r1 = t,! p" - t) . :,i * - oi =
b-3xffi=,' evezlanusin aparaq: x=o olduqda ,=2, *=l olduqda r=r
,=*i!L=J*€,,t-€f l=it'-3 |. 2 r+Jtlr)
={'.f'f+)={'."'#)!190. , = acos3t,y = asrtn3r (a$oia .ri + yi = n1.HelU
x' (t) = -3arotz $int, y'ir) = 3asirr2tcost,x' z (t) + y'2 (1'1 = 9n?cosrtsrirr 2 t + 9 az s i nr t cos2 t = ga. cosr tsinr f,
\,r'-(t) + y',(t) = rj9a2cos2tsin2t = 3acostsint.
tf , r
z\u-x ' z
Bu ifadeni (2) d0sfirrunda nazaro alaq: 0 s t 5 I,
, = f:rnonr"o", at =f .f sinztat= -T-'r,1" = -f t-, - 'l
=fBu eyri koordinat oxlsflDrn her biri ile simmetrik oldutundan almmg
L = ! edadi onun *.re | -ao' oa* ki eyrinin uzunluEu
L=+'3|=ea.
1191. x = iror't,y = dsan3t, c2 = a2 - b2 (ellipsin evalyutasr)'
E lL D0sturunu t tbh edok:iz (z
x'(r) = -3 ;cosz tsint, y' (t) = 3-
u,sin' tcost,
-- a{ crv'2(1) + y'2(t) = gicosntsin't + gpsinafcos2t =
- /cos:t sin:t1 9c'sin:tcos:t, ' , ' -:= sc.sir,rcos.t(? . # )=
--#i- (q: + br - c:cos2t).
Bunu (2) dllsturunda noz.re slaq:1r2 t1',(
, = iil, rrq2 + b2 - c2cos2t sintcostdt =
=#f r* - 6zsos21d(gq527) =
7.,.2 to= : I ,i;4 V= ,r roszt d(cos1t) =
'tZabl..i s lr
= # ro' + b2 - czcos2t)l lo=
= * [,"' + b' * c)l -(a' + Ir' - c'z)i ] =
= $ [tr,,li- (zD'); I ={ 1o' - u')
1192. x=rxrsa r, y=aiaa t
IlellL r-nim o-<tan f -vs devisnesi ila (r(r)fl n(4tesi eyri boyunca
trarctstedn,0<r<1.
,i = -4*.3 r.inl, yi = 4sin3l cqr,
i,2 * 1),2 =1**o *io2l +sin6 roos2 r)=
= te sin2 rcos2 r(cosa r +"ina r)= l6sio2 r cos2 r,, [(.*',
* rn'rI -zsi,,2 r"os2 t]= te,h216es2 1,
,
r| =a(-cosr) it =as'11.t
i,2 * 1i,2 =oz(-2*"t +*r2 t +"^2 t)=zo2 (l - cor,)= 4] 3i{ !,
t=u'i,nla=ft=Y' r=o' Y=ol=
@ =zdy, t =2tt, y=i,
=zoirny.zay = - aocaoylf = -<a(-t-r)=u0
,(r -j..' ,)= ,' -2 zQ-"az x)=2"a2 x(+*2 z)t
t = Jil "n
zr J-* " 2 zr a = fii,t t -*-*6*a =
= #(*bE; * r.rl"*,. f, -"*,4): =
= jr('r',.'"fr! I
= r. f r"(. o)1193., = ok -sint) y=o(l-*.r) (ost<zr).HdIL
1191._t =a(cosr+ tsinr). .r,= a(sl?lr - rcosr) (0 < r < 2:t) (aqq gen-a)-
IIaIIt (2) dtlstunrnu tatbiq edakx't = a(-sint + si,lt + tcost) = atcost,!'t = a(cost - cost + tstnt) = atsint,x'' (t) + y'' (t) = a2 t' ,
fzd _ r z:r r2i[L=l no+,at="1 tat=a.-:-l =2an2.Jo 'o
2lo
1495. x=a(sht-t\ y = a(cht-r) (o<r<r).
flallii, = o{"nt - t\ Y, = "tn,
i2 * yr2 = oz ln\ - z"u + t + sttz r\= za2 Q - "ni = t i tnz l'
Bulan (2) diistumda nazore alaq:
1496 x=ch3r, y = rrr3l (o<t<t)Halli
i, =xn2*n, Y't =3sh2tcttt,
i' * i,' =g"nzon'rf,n\ * 'n2t)=s"n2*nz61'v =2*2ua '
r=)T,nu.tffi a,=lloarrl:r{*r)=iiw"f;l=:1"'-i.
1497. r=aq (o <9<zr) Arximod spirat.
Earli (3) rliirsturundan istifad5 edok Ona gdra
r'(e)- a,
t =\ jl +;a de-T,F *, * o, =,1 $* t a* =ftoo
= o;lo"n iaa -(.r- ;.,4)1. =
l,^ ' -rz"tT.*n).lzlat.rm-r)-n"--k, l
= 4r[G . ;+.,tr7])1.' = 4\"^t;7;. :,o1,, . J;];l)
149& r=de (rro) (o<r<a)Halllocr<a Prtioe 96re -*.9'o oldu[u almr'
r' (9) = s12''o ,
,;$ * i (e) = o2 "h
e + o2 ^2 "b
Q = o2 "^n [ * r')
349
Bunlan (3) diisturunda nszere alaq..-0
1ot = o.l'tnn2 t"ryde= !Jt+^rr.l
.1499. r = a(t + cosp).Holll Ofi qapatr oyri olrnaqla rsinp = O diiz xattins nazeren
simmetrikdir. Odur ki, o < e < zr kimi dayigir.
'' (q)= -""mP'
,2 * r'2 = o2( * 2*" 9* *2 q+ "-2 p)= 2o2 .2"s2 I = 4,t2 cos2 g
t = +aicoa?ao=urto ?le = ao6221ot sot. r = -2- [f,r . l)t+cDse \,. 2l
. . H^alll Oyrinin uzunlufunrn taprlmasr ugiin (3) diisturundan istifadaedsk. Ona g6rs
r'(9)=tPsbq-'(1+ coepf
,2197*r'2191., p2 .- * .p2 r-2 g - p2[*2o*g**"2 ?+rio2 ol -(f . cosg)z (t+coep)a (r**rqf -
_2p2(t+qep) - 2p2 _ p2(:+c"se)a (r+msef -
[-e q'2
Ett
, p? de 1 a, - i a,..i,--:=pJ-=2ol-=_,, _I*"u! 6"*3? 6cos.r
a
,=i# inteqrahnr aynca hesablayaq. Hisse-hisss inteqrallama
diisturuna gdre
,= | . au=J*! 41@s, uts. Idt- t- =dv; v = tqt\
cos-1
oldulundan
tL*.r =
ts!-li -lutYr o,=Jz-t{ o,=,tz-r*l!t ,,' *"tio 'o ",o.'t 6"o'1 6*",
, = {. 1^1*(;.;;]l; = ;(".'(,, ?)),lrll -cos-r. I J
,"*='"lrTi=EE=*-,," 8 l,-*'i /r.*'l -fr-*,f'-,t--;Il+cos-
I=](o.r,(.,4*r)
1501. r=asn3l.
Halli # qapah olmaqla g-nin Odan 3z -ye artsnasr ilo syri
koordinant baglanlrcrndan grxrr, Oziitzihii kssir vo owelki veziyyetinsqaydu. Belelikla ayrinin uzunlu$u
lr- 3,t =- i f; 1*l. ,;' (oVe = a lsn2
?rae =00
1502 t = athL (o<o<zd2
Halll
''fu]=1 -L - ', '.,_ , ,n,3'
12 (e) + r'2 (s) = a2
" I. * #= "-*,
r=p(# *r"(Jz *r)
= |' i('. *' z1', = 1(*. 1'^?).' ='f
,h'g*-l'l=2 +rn'll
351
a2ch2g=-t-( ar1,r9"1zq *r\- a2 Gn2.*t\=kh4!\ 2 2 )+fq'
2
t=iT,ffiao=iT,ff*=
= 1'{.* ;7 p= z- - ouf2n = z* - dtir = d(zr - th F)
2
(t<rsr).
?7-r= l,l,'@\*t2(oYo
ti
4ch1 ?2
150s.
alli
int€qElmda gdcn ra kegek. Onda
ahnq
t( r\,=11,,;)
5
t=l0
da=:(,)).
l; r,t .' =1, t (,- j)]'., = i(, - r'., =
= i(, -,. 1,1-' = l' - ;. ;,. t = !, . ;. # =(;. +)',
' =iG. +Y' =(*+'r'r)1, = ; +" t -! = z * itnt
tSM p={ (o<'<s).
Halll
352
,Ei r,Ti o, =i [# ", =iFt, =
- l ll5 r
=.i(,.;)',(,.;)=. i(,.;)'1,=i(;)' :=:(+-,)=i i2=f
tsos. e=i"hP ap (o< r < R).;p
Ealli bt meselonin hellinde yene bundan awelki miolda oldulukimi horakct edek:
,b)= +. b bt *'='*,n2 p= "h2p,
Rt= I
0
1506 r =t+6/, e=r-relc (o<l<r<r).EalE
, r , 2*,s2! -t l+csr-t 06,,> =t --.-=-' , "*,
t Z(x}.2 ! 2q}.2 ! 2e}.2 !'2222
- 1l. l@s- -dfdfcfi1-=-.-=-rmt.-
-dO dt dq cos, '
4"in2l"osa 1 +"*a1,2 (e) *,'2 (s7 = 6 * "* rf . -;qJ = ff ,
T-te;
-
rt = !' .p1r1* tz qrye=i,tlW;V tan* a =
00
,2*2 L r='l-L. *'
, dt =-adr =T.6 *", Z*2! i;
7
Qalqmdar:
X inuw agu.
1s07. ( t -I)t=11". +" o I zancii xstd q0vsiiniin.t
)uzunlu$unu hesablaytn.
1508. y2 =Zp, parabolasrnm
qOvsiimfrn uzunlufunugct[rmeli).
1509' jr = lnr xetti qovsiiniln
tapln.
(r1 =0{an rz=b'ya qadet)
,^"fr !1": -"-2)'|. )
tepositrdetr onun M(x,y) n6qtesine qedar
tapm ( y -i asrh olmafan daySen kimi
c^uub, ]-,itlj *PbYr i;; P'1
2p'' 2 P(rr = JJ{"n xz = JCa qeder) uzualu[unu
1il1' y=tn"*l xatti qovsiinih (rr=rdan xz-b'yee -l
tapln.
caneu: t+11n1.221510' y = ln(l - x2) xottr q6vsiiniim ( q = 0dan xz = -l -e qeclor) uarnlupunu'2
tapln.
Cavat: n: -]
qedsr) uzunlufiunu
crurb. ro"o - u-u
"o - e-o
15E& f =!<r-tl3 yannrkubikparabolasr-um, y2 =l nu.tol*tta""lina"
qalan qolsiin0n uarnlu[unu hesablayrn.
c.*b,9fl.,q-r).e[2 \t2 )
1512' 5y3 =r2 yarmkobik parabolasrnur, *2 *y2 =5 gewesi daxilinde
354
qdan q0vsiirniin uarnlulunu hesablaym.
Cavab: 44.27
Is13' 9oy2 =r(x-3a)z xatu ilgayimn uzunlulrrnu hesablaym.
Cavab: 4a",5 .
1514. Absis oxu ve y=11e65v, .p=lnsinr xstleri ila mohdud olan
egix.tli ugbucaqlardan birinin perimetrini tapm.
Canab: ! + 2h tgY = i * zlorr"D * tt2 "8 21515'
., = f J * *.rin..6 xsttinin uzrnluBunu taprn.
1516. (y-ar"sior)2 =t -12 xouinin uzunlulunu tapD.Calab: 8.
1517. v=a(r-sinr), y: a(, -c,ost) tsikloidi tizsrinda ela n<iqta taprn ki, bun<lqta tsikloidin birinci talrnr uzunlulu boyrnca l:3 nisbetmde bolsiln.
cavab.. t =4 ot,,,da x = a( 2a 4]., =
g3 [3 21" 2
lStE ..?..1(l l, . Il ], = r xottrn uzunluEunu taprn.f al \b )
gaya6. a4::+t_. x = aans3 t,y = bsinl r oldulpnu forz edin.
I519, r = oos5 r,y = a"in5I xsttin uzur ufunu taprn.
c"*u s,fr--Lt(r..6)]1597. ( r\
r = {cosr
+ tnrs i}.v = asinr trakhisinin (o,a) noqt+shden (,,y)
noqlfiino gdsr q<tvsiioiiur ,'"t'nlufunu tapm-
Cavab: ah9 .
1520, x=R(cosr+rsrnr), y=R(sint-tcost) C€wo evolventi qovsfniiLn
uautrEtnu ( tr = 0 dan ,z = z -ya qedat) aprn. )
Cavab: E-.
R.2
l52l' x=11 -2;sinr+2rccr, y=(2-t2)ca +2rsint xoti q6vs[nun
355
Cavab:2.
uamrulunu (4 =o4an tz = z -ya qsdar) tresabhym.
ca*t, 4.31522 t Fx=t', x=t-- x*i ilgeyhin uzunlufiunu tapm.
Cavab: 4J3 .
1523. p-ag Arximed spirah birinci burumunun baflanErcrndan sonunaqoder olatr q6vs[ntn uarnluSunu tapm.
c^n bt *nl r, +? . in(2, " Jl- *).ls24' pp=r hipertolik Wiraln qovsriiniiLtr (a=fa"o m=!a l4x\
" ",,ntuEunu lrsablaym.
Cayab:b;+;.1525. p = a(l + ca5p) kardioidinin u,"nluguu taprD.
1526 7@ cavab: 8a'
p = asia" + xsttinin uannluBunu tapm.'3
Cawb:1o o.21527. x _y= l,laosxb )@tinb uamlugunu taflr
Cavab: 4'
$7. HOCMLARIN HESABLANMAST
10. M?lum en kasiya gora hombin hesablannusr. Ogar cismin yhacmi vsrsa ua
,S =.t(t) [a<r<D]cismin x Nqtes tda Orotuna perprdikullar olan Dfrstrt i lla kasiyininsahxidirsa, onda hacm
bv =ls(J)e 0)
a
3
dtirturu il, hesablarur.
zo. hrlannafutatralufu
alfian cismin hacmi. Mrtsbvi fryran Ox
a<x<b',0<y<y<x)sahainin firlanmasmdan altan cismin hacmi
b1Vr= rly'G)dx (2)
ttd*taru ila hesablantr. Burafu y(x)*osilnaz birqiynoth funkbndt.
Onuni laldaa!x1b; 0</sy(r)
sahxinin Ox oxu atrafirdafrrlonmastrtan altan halqamn hacmi
357
v --*Jlyrl1'1-yr2gt)*
dtrsturu ib taplr. Burfu y{x) va yr(x)-rrunfi olmayan kasilmtz
funbiylardr.I52& Oturaca$rnur taolleri a vc 6, yan tili ", htndurluyu /, olan
gardaf,rn hocmini taPn.Holll Otxarzr$a sahesini S(x) ila iSara edok. Bu sahs yan tile
perpendikulyar olrnaqla ond,n r nrosafedadir. Eleneorar hesablama ile
X'l=4lU,r-V.hb"l"rtL' )oldulunu hpnq. Butu (1) dtistunnda nezsre alsa4 yazanq:
h ,lin =' i x,v* 4' [ 4, -")] . * &=ortr
=iW.4."11={+.*S=!a.^t1529, Ouracaq ellipslerinin yarunoxla\ A,B ve a,r, hlmdiirhiyu iso
h olan kasik konusun bacrnini tapu..6lalli Konusa lundflrliiye perpendikulyar olan yuxan oturacaqdan
r mosaffe olmaqla mtstovi keckak. Kesikdo polyuslan
"{,)=,+ff,, z2Q)=b++x
olar ellips alanq. Bu ellipsin sahasi
(3)
t <,t -,(", L- " )(o
-,#.) =
= l* * d - 2f . u, *\o -4-! *'lolar. Onda (l) dr:sturuna gOn
.( aB+bA AB a?+bA d\=fli aD +_ -dD+ ___ +_t=
\ 2 3 3 t)=!10\o +ze)+4n+u)l
I$A OEfiaEa$ s, tiiLrdffrl(iyti iss Il olan fulanan paraboloidinhecrnini taprn.
HallL Frlznet pantrl,bidin Oy onr strafnda frlaamasrndan
Y=qr2
parabolast alms, $515 gdre parabdoidin oturacag'n sahosi S-dir.Paraboloidin oturacagl sahrsi s olar gerrrs uzorinde yetlegir. Bu gevrsnilradiuunu X ile igara edsk.
bcraberliyindan
s=d,a
buraen ise
ahnr1. Q1fuy 16l,
,=4*z -12 =2 . srvr=8.-sd|
aB -Iab+ b,4 ,18-Ab-aB+ah t\+--x+- ---T-* f =
aB - Zab + bA . J. * A.B - Ab - oB. rb ;\lh -h '2'---F -T][
h/
'=.tol*
=,(*..
,=!1"^=s .A' =oH'0" H 2lo 2
L=n2a
dIs
35t
Aqapdah sathla a ahat" olunntu, cisimlerin hacmlni taPfi'a1
tSStLrl^=t;7=9v.2=!.,/b'a
IIaIE -Cisim
silindirik ssthin va miistavhin hisseleri ile ehata
olunrrugdur. Cisnin r=ro
nriistovisi ile kssilne baxaq lGsikde dfrdrucaql alu$ ' OAB w ODC
ticbucaqlarumoxgarhfrndan
#=#nisbeti alrntr, Buadan
rc=sff=a;or oruna perpendikulyar olan ixtiyari kssiyin s1r1-sabasini
s61 zt"' l-"f=;l'- ,;alanq. Bundan (I) diiatunna g6n
t
' =+i'FT = i +"'{' i)"1 5)=llo
z r r:l=-"'rl,-il'l ='f ='r""
' 'to.),,
t532.4=+r - +!; =t lellipsoid).ol b' c'
flaltl Cisnht-asiyinin m[stovisi or oxuna perpendikulyardu- Odur
ki, ellips
olur. Onun sahesi aSkardr ki,
359
yz ,-'2 -=r
["trJ't"'EI
,r.r-""1r_i)trerabedir. Hocmin i hesablayaq:
,=*j"(,iF=*[. *)_.=
= 4" - ;)- ""(- ". l)= *(T . ?= ry")).t
tsss. !-t\ -?i=r, "=t,.a' b' eLtrIaIlL Cisfun biroxlu hherboloid ve
miistavr hiseleri ile ehata otunmugliurl'a*turo nq"tr, xol musbvisinasimmetrik oldulundan ,uxan hissanin hacrnini hisablam aq (o < z < c) vealums naticeni ikiye vurmaq lazmdrr. Cisim
z=cb ("r.")miisavisi ile kasilir. Odur ka ellips
,2 -. y2 =,(f))2r[--T\2
l'i'.il ['r/'.;;1olur. A,gkardr ki, en kasiyin sahssi z-in furksiyasr olur.
"t,l=-r[,-4]t .'/(Xur kr
, =, *il. iY = ^'(,. *)llo= 2 - o ! = \ *,"
1534. x2 +22 =o2, y2 *12 =o2.
Halll Crsmln I ni.r*io. baxaq. Cismin mtstevi kasiyi oz8perpendikulyardf. Onda kvadrat alrrq. Onun sahesi
s(z)= o2 - "2
'Bu halda cismin hocmi
, =*(" -,)*={,',-Il. ={"' u)=+
t535. rz +y2 *r2 --o2, '2 *Y2=*'IIallL CAim
'2 *Y2 =*
silindr sethimn hissesi ve
,2 *y2 *12 =12
sferasrmn hissesi ila ehara olunmu;dur. Odu ki, cismin hissecikltri xolaiirt"ri.ioa oozrren simmetrikdir. Onun 0lz<a yuxan hissesina bapq'
Miistavi kasiyi oI oorDa perpendikulyardrr' Odur ki' eyrixotli trapesip
avedamasini a[nraq:
P=o olduqda r=o'| ----; t ,
| =,1*-r' otduqda ,= Ercsh{;
."*-E ."*,E.r,,,=r-"'-l=f oz - *z\*,2 at =(oz -,2\ l"'fr*"*uY,=o\Lt-L
i \- - )- \ / 6
=(,2 -r2Yur"rtn,E..E).\ /\ \a+x a+rlAxrmcr ifadani hecm dushrnurda nezere alaq:
,=rj1,'-";[,,*-8',*0
Bu inteqrah hesablamaq ugtu x=argze
361
almq. Onun sahasit'-j ----------- '[;7'V(r-r l^ /'ls(r)= '-j ,lo'-l'" -v'W=2 j
T-- 2t o
=(,'-rl,.i,-,)1""""r;{; =[_*i,
I ' I -J I r lll- + -stnz aTcsrn\o+x 2 \ Ia+rJl
evedcmesi[i aparaq.
6=2o,r9.J" 4q,cns'Q
r=0 otduqcla e=O,
x=o olduqda e=!.4
o2 - *2 =o2 -o21gag=ozfi -Eagl
r-; ,lG =,,1!3:g =o,a2 e.tse=s_pooe.a+r a+ag2g
Bu oCicalori hacrrin dustrunda nszere alaq:!
v = zjni - E' qh*sin(sinp)+ sinpcose] . 2o ts? aq =o_w,e
= o s itr -,r' *\r, ffi),r,o t r *l - *'h 4+d - @l.[tL 2 6 ]
.itr,, - u, -v,l= *f* *y". :,r, *. - :io, *, -
.!t',-o",b,]=^,,1.r.:,r,*,-:i,-,",)=*yi-i*,-.\i-
- 2(+ - *. -, - 4l'"]=
*1r.i ; - :. * - :. ;{) =
,:{r 2\ 2l 4\-'" lE-;1=1'1"--,1$7, zz =bb-x\ ,2 *y2 =*Halll Cisim
*2 *y2 =*silindrik ssttrin hissesi ve parabolik silndrls
362
z2 =b?-r\ehaia olunmugdur. Cismin yu<an hissssina (o < , < '[a) ba:oq' Cisim xormfistovisine simmetrikdfu, 0x o:ama iss perpendikulyardr' Odur ki, ssthi
E:? _l; 7s(r) = 2 lrdv =2Jo("'r) ldv =z'!br\a-xl
00inteqmt ile iftda olunur.
Cismin ndqtalarinin XOf mii,stevisim simmetrikliyini Irazar3 alarEq
hacmini tapaq:
-( \ 1l (^ t ^ 1]|"v =4Jbi,l;("-xE=rliil-z -,2l*=dl+" -;,'ll =o ol. .l l,' '
)1,
=^aftuF tt{,r)l;r*.1
tsn *+*=r, (ocz<a).ol z'
Hotll Cismrn kasiyi oz oruna perpendikullardr. Ellipsin o vo zyanmoxlandr. Odur h, sethi
s\z)= nolur. llecmi iss
1539. x+y+22 =l; 1=Q; Y=Q; 7=0
Ilalli Cisim biriaci oktantda yerloqir. O koordinant mfisovileri vo
z = ,tt_t_ y
sethi ils ehaa olunmuqdur. Cismin mu*cvi kcsiyi oz oxurrperpeadlkutyardr. Odur ki, o beraberyanh diizbucaqh lgbucaqdr vs
l@tetbri (l - 22)-Da beraberdir. onda
s(z)= lf r-22')2'-2\.' - /'Bu halda hecm
-a9 ztl arY=frlzdz=tu-l =-' 1t ao _lo
, =!l(, - z,z *,a\a, = ! (, - ?,1 * 1,s lr = I f r _ ? ..1) = l. Li - t0 t 3 - 426\ r 2\ 3 5 ilo 2\ 3 5) 2 15 15'
1540. x2 +y2 +zz +r!+).tz+zc=a2Halli Tenliyi y + nazaran hall edek :
y2 +(** ")y+
12 * 12 + o-o2 =o,
-('*,1*J+] -tr2 -* -zo.=--_--2i-'
ir2 + 2z + 322 - 4o2 <oSortirdrn istifrde edarok, r-r tapaq:
--- rt,[? -r] *rz] -r*tFz] -u2 -"rzJJ t"2'=
--:
.-- = 3
yeni
,*zJt]lz? ,,.26,2 -zr2 -,33
oldu$mdaa alanq:6E-lraszslra.
Oz oxum perperdikulyar koeiyin sahasi
2.,1;o2 4 -"s(,)= 1 ,8", -rS -t? -r*,=
-44e-:j3
2,,[;,\] -, ,I I 'l:= i I qo2 -t 12 -t( "*'-\- lt a,
zJr] -zrz ',1 3 \ tl l
-3zJJii =eG) isaroetseh
,t(")_,
,(,)=+ i I e("\-(t,*,121)a,Jr A(r)+ rL I-3
3x + z= A(z)smt evazlemesim aparaq. Onda
364
,,
o,)=#jf'*=#l'.i*")l:tlecmi tapaq:
o' 1r1 2r{v' - zr')-\'=--)-'...:4,- oJl 3J3
ASafukt xatlorin firlannnsYfun afuan sathlarlo ahte olunmas
cisi,nlorln h* ini taPm:2
tslt. "
-- t(:\1 (o<,<o) a onr atrafnda (n'aloid)'' \a)
Ilarri Cismin heqdni
v,=,!t2e0
dUsturu ile tapaq.
,.=,iu j.'-=:'4,l51Z y=2t-x2' Y=o
l*'li!
a3
!,.b2
a) e oxu .trafinda; b) @ ocu fafirda'irUi
^\ * oxu ils kasigms niiqtesinin koordioantlanm apaq:
2x-x2 =O>t=O, x=2, 03 r!2
2 2, 2'
,, =,i r' t =,ib, -,'l a =,1Q,' - 4r3 + xa fu 'oo0
=,(1,' -,n * ls), =,(?. ? -'.) =,*(3 . 3 -,) =,0"r
0 *fi's = f,
6t ^tE z3lf- *- 'lz-1'r'
a1 r, = z,l.Q., -,rp.. = z,j(zs - sfu =
=2"fts -!t)',=,4!_f)=* " (i_i)="" i=y
1543. y = s11ty; l=O (O<x<z)a) Or oxr atmfinda; b) e, oxu etrafinda.
HruL ^)
yx = ri da, * = !"! 1r _ *" zr4, = i(, _ ! * r_\1" = 4_o to 2i , )lo z'
b)
v. =2r*tx" r* =l'= '' dv = snxzf.'l -, o Vx=da, v=-cos, l-( -t \
= 2rl.-rcosqf + lcoa*)=zrlt' +";"a[)= z*
1s44. y=fr)'z, ,=rl;la) or oxu etrafinda; b) or oxu otrafinda.
Halll o5zi1.onn hosigme n<iqtolariain koordinantlanm apaq:
,(r'=rL.i-(r'=[|,r =o; 14=r,r, x2.3=+a, tr<aolduqda
=,-(* ilr=z^,r() il=*
1545, y = e-', r=o (o<,<+-)a) or oxu etrafinda; b) oy oxr atrafinda.
(;)"l;l(q, onrna nezoon simmetrikdir):
^t,. =, *,!(! _
!_b = *,(* _ *)l =, *, (: _ :) = ry:
b)
,, =,*i,(; _ $),, =,*!(* i)* =
Hattlal v,=,'f" ua,--i"'l|"' =i.b)
v ' =2t I xe'^ dr =l, 'o E=dr, ,=-.-, I --( ro 6 t,)=
= zo w ( - *"-,le - u,1A)=2, tn (- *-A - "'A * r)=z,r-
,4++-l r0 ro / r-++o
1546. rz +(y' tf =a2, (o<o<b) Or oxu strafirda'
II?llL o, oxu otrafilda flrlanruadan abnan cisim merkczi (o'a)
.oqt ti"L oUo srtre ile ohata olunub' Bunun yuxan hissasinin anliyi
(r = D diirz xgttine nezeren)
Ya = b + Ja2 - t2 '
tcnliyini y *r"raohall tdrk'
t ]- 'j 12 - 4r2 +4oz
a$ag hissalinil: tediyi ise
Y' =o-J7jolar. Ax-tanlan cr"smin hscmi
,. =. iy| - r'nb =s*iJ7 -7 * =
-lr=asinr, [o=,.i)l=r-rrf"*ra, =-l14, = oeos tdt 0
\= t -2 ti(t + cos zt\y't = a mz t(, * |"in,,1' = ztz a2 b.
o\r)lo1547. rz - ry + vz -- a2 or oxu frafinda'
Ealllt2'rY+Y2 -a2 =o
,=!'2bu cismi ahata edan fiqurun simmetiya oxudu'
,114(, - t' =1, Fi-7 .2v 4
lzt 2o I
" ='![l;- F - -F)' - f.1; PqY
l=24 2q-E/ \ -El
"
-\2
=r'il,' - *.,1"' -i,, )d, - r,ti
l.' - t -,p4
)- * =
t . ^,2 ,p ^,4 ,,, f=,1# *f .ie-14;1, -'4*.'.*ff .i1u ;r),ll;l=
2a3(s I l\ 8az3=_t _+_+-t=_3 \2 3 6) 3
154& y = e' x ^lil;, (o < r < rc) Or oxu otfafind!.
Halllsinr>O (2krsx<*+2kr t =0,t2,...)
+6 E +2ttVr=t 2 I c-usin t'b =0 2kr
x-?.bt =t evsdemssi aparaq:
yr=rE "-4h.1"-2t ibrdt = r*! "-+ko"-2'(.r.rrrr"r]'='=,t=0 0 i=o ) '1,=n
=r-(1a"-2r\+f "-tkr -r.l+e-2r -_ r5\ )*1o 51-"-1r S(t_"-2r\\,/1549, x=q{l-siat; y="(r-coer) (o<t<zr\ y=o
a) Or oru strafind4b) 01, oru etsafnd4v) y = zo friz xfii otrafinda
Halll al llccmi
v' - o'f Y2 a'0
d0stl'u ilo tapaq,
36r
2t ' ^ -t2i(t-r. r*:.o"2r-*"3r)ar=vr=, ! o5(l-"*t)z(t-c"stPr= . ,, )
= -r'f (, - r.ou, * !, 1 * z,)a, - -3'f 1, - *',yt*,1 =
= *3(1, +,;,, *10* r)" - -'i*, - i*',)l' =
= p3 .1.2, =5r2 o1
b) IIecmi2t 2r
v, = z -' io*W : io a3 (t - s;a t\t - oo*f a r =
=z#zf t, -"ur{r-2"*r ,*"2 rp =o\
= z#21 (, -u**rcm2 t - si! t - 2sinto*, - "o.2
r rirr)d =ot )
=z#2f (, -a*"r+{+ lrcos2r -sinr -rlozr)a +-- 6\ 2 2 )
* z#,f *,2,a 1*",) = r;li +t. .* rA" . ]* r{'] -
- +-32f, * d, * -tzf, "* Mr * z-3 .
*r''12* = r-' 1r rz1 -o o llo \
- 4ru3rr+ u3lr=613o3 - 4-111* -?12'
,, =' i, * * =ff !;,. i = ff l'1 =,"",fi' -'i"a* = "* 4f;" = o.
Zr l=u, ly =c,sz,dt,l , 12r 12t t tl,.rr= rt.*zatt=lrr=au. ,=lsaul=;'rrt -i lsinur=icost[ =o
Bunlan yu>caoda nszem alsaq:
y 11| = bt- a-
v) Yeni koordimft sistemina kegak:Yl=Y-24, xl=)r,
onda ax0anlan hecmv =ti-v2
olar. fi -hii'ndiirliiyii 2za, radiusu 2rl ole dairovi silindrin hecmi, rur -isa2rE - .-
vr=r j (vrf ax,0
t'l = 2@ r4a2 =87c2 a3
v, = 12 f lo2 1r - *s tf - a o2 (r - oo" t) * t o2 | 0 - *" ty, = rz o3
Yekuo olaraq:
V =Br2a3 -o2o1 =7o2a3 .
1550. t = asinj t, t=bqe3 r, (o<ts2r)a) o, oxu atrafinda;b) oy oxu etrafinda.
Halll r-nn o-h ! arasrnda doyigmosi ile -t o -dan o-ya qsdor arhr.2
Venlcn ayri qapaft vo koordinaat oxlaflE ftrzeft)n simmetrik ay,rilordir. t-nin o-n. zt arasrnda dsyigmssi ile hersko edan ndqte (o,a) ndqtcsindensaa oqrebi istiqamatinds bfltiiLn syri boyunca hereker edir.
a),t 1t
2. r
vx = 2r ! y'dx =ldx =lasn2 tcosfii= 2ri (a*"3 ,\2 .u"n2 tcostdt =0 , ' 6\ I
tt;-
=e-r2l *"1 rr 2 tdr = 6nb2l ( * .7 , - *"s ,\a, = o*2 ,0 0\ /
t
' f f[,-,.2,)3a1,,4-f r -,o,2,]aa1 ,a,1]=o*r2 *oL\ ./ \ ./ 'lr2r
' f I t -tsin2 r. rsin4t -sin6r- I + 4sin2l -0\
170
- 6sin4 r + 4siu6 r - t",8 l)a("ro1)=
t-:
= e ^r,2
tr ( su-2 t - lsi.o4 r + 3sin6 r - sin8, la("u r)= r-a 2 *'-- 6\ )
lr r. 3- s..3.-7,-!a,,e ,\L =r*z(l-l*l-'),[i*,r_-sin,r tisin, e ,,lo \r 5 7 ,,1=
,105-189+135-15 ., 16 32,0fi2--o^t' ----fii- = oaza- rtS = t05
b)El
v.,=<rf o" 3r'tr*3at=e*tf ur3 r ' cos 3 r ' 34 sin 2 l costdt =ti)o,tt
= t:-2a? rio5,. "* 7 tdt = -12 a2 b,J l.r - *" 2 r')2 *ro a1-" 11=
o 0\ /E:
= -ttnztj ( r a t- 2*r6, + "o.8
r),/1.os r)=6\ )
E
= - rt -2 I L *rs t -2.o.2, * l"* s r) z -- tz*2 t( !- 2 +
I ) - 32'E2D
'- - -\5--- 1-- n )lo \s 7 e) lo5
1551. t =2t - 12. y=lr-13 eyrisinin
a) or oru; b) oy oxu atrafinda firlalmasmdan ahnan cismin hscnrini
tapllr.Halli.
x=0, y=oolduqda o<r < z olur. r -nin o dan t-e artmasr ile, xda o {an l'e arbr. l-nin t den 2 -a artnasr ila, .t t {an o*a azalr. Odur ki,a)
r, = - *l y, * -,j yz 5, = -,f y2 * = p, = (z - ztpl = -,1 Q t - fl zfi - tp t =
1000
=-z,ffrat2 -aa *tu\r-,y,=-z,l(e,2 -8r4 +16 - 1613 +8rs -,t\t =
=-2,(t4,t -9f ,!r1-!!ra * 116 -J,t\l' =-rf,21 -zt *27 -ro *zt -rs)-\3 s 7 4 6 r llo --(3 s 7 - 3 - )
= -2,. 21,( \ - | * ! * l) * z, . N - -zr . 27 . ! + zr . ec = zn. LB:r;r, - ttzs\ = 91\3 5 7 31 35 -" 35'---' "--' 3s
b)
v, = -z rf 4a, - z,ci * = -rrib, -,rfr, - rrlp - ry, =100
at=erilu -# -2ta +tztt - ar'h=Y
o'lo5Polyr knordinant slsteminde veilniS sahal*tn firlotnasufon
aLrl@ cisimlein hacmini tapw:1552. r = ai + cas p\ (o < 9 s zz):
a) polyar ox etrafinda;
h) r "os 9 = -i diia xesi atsafinrlo ;
Ealli
, =,41,r6y*"6, (r)3;diisfinndan istifid. €dak.
a) Polyar or milstrvi fiquru
o<r <o(t+w9), o<e<Zr
iki braber hbseye ayrn. Ona g0n inteqrallamam odan E -yo aFraq:
y =z]i_j1r*,nef ,auo=J0
= -'!i7,. r* r+ ros2 e + cos 3
e)d(cos e)=
z-3(.---.3 .2-. , f*"or.1|0 ==: l*9*r*-9+@s-9+- A
=,pt f(, *1 *, *l)-(-, *1 - r * l)'l=d.o = s-3
3 L\ 2 4) \ 2 4)l 3 3
b) -, =y, yr=-,-I diishrrlan .ile yeni koordinadara keqak. Fiqurun
Orlr oxltlta nezel'n simmetrikliyini neara alsaq:
4
v,,=2* t\,,\re,
Burada eyrinin parasretrik onliyinden istifide edak. Pararnetr ohraq g-ni
got-urak. g-nin o{an {-+ qader artrnasr ile r, -in odan 3f-a qsaar
aralrusr nazare almmrgdr:
v o = z#il(r * o, o1", *!)' (*, * z *"' e - rls.
Inteqralaltr ifrdodo cos p-nin tek dersccsi qtirak edir; beb ki,
,l*'r*'d'=o'o
Bu hedleri nezarc almasaq, alanq:
no =r*o'i(o-o,+2*6 q-"o"2 slq =0
L2, - \= ut3 tpcosa p +z*"u p - *"2 s\p (t)
0
Burada daoccni a9a!1 salmaqla hcr bir irteqrah hesabbpq:
f o*"a c*=1(z*zr)2 a*=10**"zqPae=
,f
= ? (, *, *, r, * "^' u), = !(l., *" r*. f,* o r) * =
=(Jr*,,rr*]o -)f =+
lt
rEll-^;1 z*"6 dr=t
21( z*r2 q\] ae= t,l I , *"zef ae=
d 4ot / qo
,t:=\1(t rt*"ze * t"*2 ze*"*3 ze\ae=! ,4ir, ') 4
rE
,'1(, ,t*"zq*3-^ *] uo,glae* l?[,-,-' ze\oosz@e=o\ 2 2 / +o\
Ett-
= ! 1 ( t, r"*rr. 1 -" qo\q r_l ? 1, r -,,n2 ze)a6,,,ze) =46\2 2 ) E6\ t
,t ,f
= I(!;*. l"*r*,i'-r)j * ]['*ze - ]''3 z.)f = ?;rtE2 ., t2 ., r( I - \ll rI cosz s19=;rll+aszfldq=: .9.: "iorr)l; = ;
Bunlrn (l)-de nazars alsag alanq:
v- - t-411 +\-!\= -'(zr, tE) -13.-2a\
Is$. G'1 * r'f = o'(,' - f)a) or oxu atrafinda;b) oy oru etrafinda;
v) y = .t diiz xetti otrafinda. Gtistariq: Polyar koondinantlara keqin,
cavab: a) r, = yll ar,(, *,6) - ?.], ul rr" = tl, 9 z'-i4L ' ',3l' ' 4.12 4
Qclqmatar:
Cismlaria hacmlarinin haahlonrnast,1554. y2=4, parabolasrun oz oxu otrafinda fidaomasrndan ahnan sath
(frrlanma paraboloidi) vo onun oxuna perpendikulyar olub parabolarun
aposinden vahide barabsr mesafede yerlegen mustevi ila mehdud olart
cismin hacmini hesablayrn' cavab'. 2tt ,
1555. Boyuk onr 2a, kigllr om 2D olan ellips: 1) b<iyuk ororn otrafinda
frrlanu, 2) logik oxun etrafinda firlanr. Ahnan fulanmaellipsoidlerinin hacmini tapu. Xtisusi halda hi'ranil hscmini alm.
Cavab: l) !, obz . 2l !- o ozh .'3 3
1556 Oturaca& 4, hiindii,rliiyu , olan simmetrik parabola rqmcntioturacagr otrafinda fulamr. Bu halda aluun firlanma cisminin(Kavalyeri "limonu") hacmim hesablaym.
Cavab. lt hza .
l51557' *2 -y2 =02 hiperbohsr ve x=a+h (1,>0) d[z xeti ila mohdud
olan fiqur absis oxu etrafinda firlamr. Frrlanma cisminin hocmini
tapm.
,rr5. otllo*t).J
l55E y=xex xetti vs x=L .v=0 fruz xotlari ilo mahdud olan aFixotlitrapesiya absis oru etrafinda 6rlanr. Bu halda ahhan cismin hacminiapm.
c.avab: X@2
-t).
1559. e* + e-t, ='-? zoncir x.fii absis oxu otrafinda firlaur. Bu halda
katenoid adlanzn sath ahnrr. Krrcnoid vo baSlanErcdan a ve D vahidmcsafrde olub, absis oxuna perpendikulyar iki mtlstavi ile meMudolan cismin hscmini tapm,
czvab !1"2h -e-\b -"!-4*216 ,114L 2 2 l1560' y = x2 va y2 = r parabolalarmn qdvslari ilo mshdud olan fiqur absis
om otafinda firl"nrr. Bu halda aLnan cismin hacmini hesablayrn.
Cavab,L.l0
156l' y=2r-x2 parabolasr vs absis oxu ilo mehdud olan fiqurun ordlnat
o;or strafinda fulafinasmdan alman cismin hacrnini hesablayn.
315
CawaU: fIfiZ / = sin x sinusoirlinin yaflm perioduna uyEuo qdvsii ile mebdud olan
eyrixotli trapesiyann ordinat oxu otafnda fulanmasrndan almancismin hocmini hesablayrn.
Cavab: 2tt2 .
1640. r=a(r-sitrr), ,=a(l-cosr) tsikloidinin bir taF 62 ohmcagreFa.firrda f rlanr. Ahnmrg sath ile mahdud olan cismir hacninihesablayrn.
Cawb'. 5r2a2 .
1563. Tsikloidinin taSr ve onun otuacagr ile mohdud olan fiqur, otrracapnortasDa p€rpendikulyar dilz rcttin (sinmetriya oxu) strafnda fulamr.Bu halda alman cismin hecmini taprn.
cu"ur,"rrfd-gj.( 2 3)I 564' ,2 - 2"o-2x )6tti 62 asimptotrr etafinda fulamr. Bu fulanma
nsticasindo alman seth ils mehdud olaa cismin hecrnini taprn.
cawb:ff.I56s' y=e''z lorii vo omur asimptotu ile mahdud olan fiqur ordimt oxu
strafinda fu16rrr1. Alman cismin hacmini taprn.6 2 .l=
Cavab: z. Burada Je-'-as=f (Puasson im&qmlr) oldu[nrndan0
istihde ediLno lidir.Is66 y=e''z wtti ve onun u$imptotu ile mehdud olan fiqw absis oxu
etrafiada firlamr. Ahnan cismin hecmini taprn.
Canab: ,f;.Burada ir-"a= f fn^*" ineqrar) oHu[undan
istifade edilmo lidir.1567' y = y2"-x2 :ottiain oz asimptcfir otrafinda firlannrasrndan alnmrg sstl
ile mohdud olan cismin haonini hesablayn.
Cu*u' 3"rP nu"aaa *lu" *=I (puasson inteqraL) oldulurdan3262
i$ifado edilmalidir.
t 56E" ,=lll xetri va absis oxu ila mahdud ohn fiqur absis oxu etafindax
fulanr. Alnan cismin hscmini hesablaym-
Cauab. ,2 . Buada TY&= i <oitl-t" inteqral) oldulundan
istifrda edilmolidir'
1569' f ={;@ro) tsissoidinm 62 asrmptotu otrafinda firlanmasmdan
almmrl seth ib nehdud olan cismin hecmini tapm
Cauzb: Zii . *= u"u,z t, r=48J qabul edarsk paranetrik
ifa&Ye kogin'
t 570' i
. tO . '; = ,ellipsoidi ile mehdrd olan cismin hxmini hesablayrn'
Ca,,zb: !d". ,=]1t,X dilstunnu tstbiq edin' burada sft) en
J"t
kosiyinin sahesidit.
t57I' ,=+-+ eltptik parabolas t ve z=r mtistavisi ila rnahdud olao
cismin hscmhi hesablaYm'
1572 *2 *v2 ' rz 't biroyuqlu hiperboloidi v2 7=-l'7=249
Czvzlo'. r Jlmiistovrleri ils
mchdud olan cismio hscminiaPn'
I 573, , = "2 + 2y2 panboloidi ve x2 + 2y2 + z2 = 6
olan cismlann hecmlerini hesablayu'
Cavab: q =o't\<z'[e -\1,
Cavab: $*etlipoidi ila mehdud
u, = oJlpJa*rlS.
1s74' *-t. -*=, ikiovuqlu hiperboloidi rle rr.+-+='349
eilipsoidinin kasigmosindan ahnan cisimlarin lrscmlarini tapm'
Cavab: '1
='3 =qr(Ja+J1-g' vz=tr('{3)'1575. Kvadra"En markezi a radiuslu &irenin diameri boyrrrca harekd edir'
bu halda lcv"adrmn yerlefdyl mflstavi dairs .mustavisina.perpendikulyar ohraq qalrr ve krradraUn iki qarqt faposl gevl'o llzftl
377
hsrakat edir. Herekst eden bu kvadratrn emelahecmini tapm,
gotirdiy cisndn
Cavab: 9a3.3
s 8. FIRLANMADAN ALTNAN SOTH SAHaSiMNI{ESABLANMASI
- AB llE,rr, eyrisinin or oxu otrafinda firlaumasrndan ahnan sathinsahesi
p =zol,;,atdtisturu ilo hesablamr. Burada ar qtivsin diferensiahdrr.
Asapdah ayrilann firlanmasmdan alrnan s h sahalarini hesabhyn:1576, y=r.f (o<r<a) o, oxu otrafinda.
Halll a = ,! * y,, GY, oldu[undan alanq:
,u,=f4l=+,i\,2 ) z"z
dl = .h +9t d,\4a
e = z,j r6lff;,a. = z,i, f . fifa" =.-. 1
=+[xr..,ta;
,*?!=t eezlemesini aparaq, f;.r.f ua.
F@,,=+,,-T,,,
378
Burada
llo lla, = + Ll +',Ffi ,, =+IF-g)'.
sl-lla
ZrrT-;- Jr 2.r,
,
lla9Ir= J
Zo
Tlla
il,'r';t'111" {i+ +;)' 1* +t1l=9
l1a 1
, Fr6 . = :ill' l+)')"[" - t+)' J
=
9
3_rlrrzor)i r o1''[if -o3 rrz'6 rr -l'g rl'/i3 -=l[
" J =3 --F-=--;{- e3
o1 .:rrJr
12 -ifiteqralnm h€sablanmasmda malu'ln
1,f" -,' a = tG] -!r'1, * J,' -.'l
*r .t,' ^14]=ttJtltl-
4 "tt*lr'i sr -'l gi s4 8l 2
Bu naticsleri p -nin ifadosindo nezere alaq:
l.,t ort3.,/iJ znl *!1o2 22,-tt*rri-r l-,=; " .;i-._;L? _ ,r ,_z--J=
tt"!t-l*z rltt{tloz to2, rr*rJrll--ii''ii- " "-' l=
llal8
= *l"tt - ry r,"'ri")= *f ,,u, .'u{" ]
- .- ,z i
-1.{znlr"lrrz (s/ I',=i;,FH
llo
,'.6il;9
dristurundan istiftde edsk. Onda
lSV. r = "cosfi l14st) o, om strafnda.
Halliy'61= -r.Igo3= -4rioE2b 2b 2b
p ]@T ,tr. ,,a,,i)Axrmcr imeqrah tapnaq iigtn
cos
&2b
{[](;-;,),(#"#)=B2b
+bP=b I a
-bItaI_$nl2h
I
I
i5aro asek.
, )Esltr- _dr
2b
,= O I
Ifr2bt
tot2 |
!,F t.' a, = ;J7;7 . +n1. t ."ErJl
diisurundan isti.6da edek. Onda
,=*l;a;t -;4.r,ri]f =
-+'l-'; L*
157& y=ryr fo.r=I') or oxu otrafitrda.\ 4)
aeu a=^f,*-;*,I c6'.r
4P=bt6x
0,F+I COs r
a(.,g',)=
F#.;"8.f#fi
=,"i'M-="?W
=f+ tgzr= t = r =t, t =2, t<t <Zl='j+" =
"iH"+fi#)te. .,mrl=OJ
= "["{;.,Q)1' . vs - o] = "[tvs -
o)' r"Lt-arE:i]
1579. y2 = 2pt (o<,<rs):a) or oru etra.6nda;
b) D oru etrafrnda'
EalL,O'=rO-r'=L=j:*
,. = r. i i rla, = r,] O - [-* t-* = "i,'r* ff 1* =
00
=2r,6\ pr* pa'-lJzt* =, avazlemsiaparaql= zrrfi '0
,Fp,,* =r, li ",\.t* =r11lt,. ro{ o+ r* _ o,}JL
J P 'lP
-E; t .2 ,,1
r, - i " f 'li11$t.1'' OV + =f = h''' b'l = 1= 4t,0
'T *Ffu=4ff "G'.rY'=?"
"lr,t, r,rP - 4l,,F,i . o, ^(,
* ff * r.t$6
nll
381
= #{^@ ., ^f - {y,y;;g;.-an],. o,,
" ^aa: l4'*\=
il.o * ravzaa. zn) - r ^Jri
.#.i)
tseo.4*4=, (o<D<a)
a) a oxu etra6nda;b) oy onr etrafiada.
Ilali Ellipsin tanlilni parametrik gakiltls faz:q.,=acoar, y =bsint (O<t<2l'\a = .t / 91+ y\ryt = ,[", "i", u t, *,J ,a,
a) Oyininn q/ oruna nozoon simmetrik oldu!.unu nazsre alaq:n,
e, =nrl rpytlrl= oa'1"-r^F "ioz
u o2 *J ,* =00t
=+al"io,,to, -@ -rrEJ,rra, =0
0 r-_:-=art I,,la, - 2 - l"!* 2-ra(*r r; = o* ri ff 7 "I
burad
ellipsin eksonfisetidir. Axrnncr inteqralda
z=ry6
svadcmesini apaaq. Onda alanqt
^ 4nb*F' 2tuhd+.eI cos- udt = --- lll + oos2uVu =e 6 a ['
= ?Q . ;"-r)] " = ?6-'-" *
"vr- ", )=2ab( b\ 2tuh
= -r= f *,, *, -;)= -tf uru;o, * 2*'
382
b) Ellipsin O-t o:s.rna mzoron siurmetrikliyini nezerB alaq:
I
= aruj,jt' +Q2 - t'1 P2 * = aa2elO0
h2_;_-;+z.dz=d'-b'
- r-, "1,.F;8. i,. r,. ifFll -
l, a'e' a'c' a'e'
]1,
='*' "ll. f,"i{' * "i1='-' * ?'41,"[;t' -')]
15t1, '1 +y1 =d1' or oxu atrafinda.
Ilali Astroidanm parametrik tanliyi, = o *"3 t, y =osit3 t (o st szr)
pHindedir. A,{kardr ki,
at = {-- 3o*"' rrio r[ * $r ru' r"o" r]ar =
= sr* rriorJ fiiE ndt = 3d cos rsitr rd,
kimi taprlt. O1'rmin ntiqtolerinin koordirant oxlan or,simmetrikdr, Odur ki, alanq;
21c = e,rlt(W =12n2 fsilr-'
t cos tdt =o0
o|,-o lx ztira
!2
=l2zz2Istua tdktnt\ = l2azJ0
1562,=aehl (rl< D):a
a) or oxu atrafinda,b) O oxu otrafioda.
UaUt alat=,f,*ya1rP=
.L*',11 =Y-')ro)
3E3
l+ snl !4r.,
P, =2r I yt)dt =2tu I
D y = acnlbdiyini batqa sekil de yaz4.
!-t2xy eq+e a .2y eg +l.=
-=,-
=-a
ea
S*,t*=r-i,
zxxun7 -r*"o *o=o-"1 -Y*JY2 -o2
,a
| =rv+,[7- ] -, =o^r:-{7},daor- f,^ 2
,,tu,=o2.---L:,,1Y' -"" =
v *',1Y2 - o2 o
"hrat
b
I-b
ch2 !tu =,t
z -i(r . " n 4\y, =, *(^ .
; "';)l:" = z -(o . i "r,'z b
)
*h: f z -zpy=ztu !" 1+:-.rY +
'lY: -o' 4=" .,ly' - a' u
t ,P-u,2 *Y o;w-F=-=w
.vdvl, dv=-i+=),ly'-a" ).=1__,
o<y<nh!a
ri -;
3t4
.17:*b,-JV-"l*"': -. ollbl=z-^')
( . *t Lrar! )=z-lrshlb q o - o"1ro- 1o l=2s*
[" ., a
)
,l*t b-u("t b-. aL)-.ab-,.f=z-1". *nb-- "a,b-).
1Skl r=a(t+q,sp) polyar or gea$nda.
HaUL (o <e< t),
,=73;a9=a{t+cos q)si,,q= q.4*r2I " !o*Er=
+or-e*"3 !,a=,[716s*tn*e= ,21t* *9f + a2 sioz g9=
= ,lo2 * 2o2 *9 * o2 *"2 9 * o2 i.2 gg=
= E.r;# *, d, = "$. F"# ?e = zo asla oB'mlrn dusulda rE zero alaq:
p = z,i y@Vt (s) = rc -zi *"a \,*l a e =00
= -16-2 ,2't"osa ga( ,*p.\= - 32n2 . L*t y'' =32 -, .6 2 \ 2) 5 2lo 5
S 9. MOMENTLARIN MSABLANMASI. AGIRLIQ MORKOZININKOORDINA}ITLARI
t0. lvr r-maddi noqaler sisterni olsun. Maddi ndqtelar sisteminin
kfilesi z, (i =12,...,n) olu. y,- M ,tna i ncirqtssmin ordinantr olsun.
,,=fr,r,3t5
(_.11""'i-*^*rur"Fi=
_q
".rg lA. .)=
komiryeti bu maddi nOqtoltr sisteminin O: oxuna nezaren statistikmomcnti olar.
'.=L,,tise mrddi n6qtalor sisteminin or oxuftl nezersn etalet momenti adlamr.
Tutaq ki Daberiy=16)
hama alrisi boyunca sathi srxl$
olao miintazem maddi n6qto paylanmrgdr. Bu halda (a i .t < r) hissodekoordnart odarma nazeron statistik momertler
u, =01y1,1{* ya a, *, =l-rl * Y 0,.
Koordfuant oxlaftra nezoran atalot momentlerib. ---i b.r
^l, = lyz (x\tl + y'a dr. ly = !rz.lt + y'z dxi, a,
diistrrlan ile teyin olunur.y = f(,)
eyrisirin koordirnnt oxlanna rnzsren xatti srxh$r
oldu{a a[rrhq raarkezinin koordinantlan
,=Y, M-,l=tkimi taprlu. t -t = f G) eyrisinin (a <.t<D) uzunlupudru. Deyak ki, oroxu oyri ils kesilmir. Onda
l,tp=W).Hor tcrsfin 2r -yE vursaq, Quldinin I+i teoremini atanq.
20 . Birclnsli ayrixatli trapesi]rnmy = 1Q), x.a, r=b, aax!b
(soth srxh$ emsalrtG,y)=r
olrnaqla her ycro barabor paylanb) koordinant oxlanna nezpren statistlkmomcnti
bMy=s$l(x)JfGW
o
dtishrlan ila, koordinant oxlanna nazersn hemin oyrixotli trapesiyanrnatalet mcrnenti ise
u,=w@lf 6y,,
r=Yv. ,=Y-,sdiishrlan ile tap lr. s -trapesiyanrn sa]rasidir,
158r'. Radiusu o -ya baraber olan yanmgevreoin qgvs[niln uclarmdan
kegen diametra nezaren statistik nomentini ve atalat momentini tapm.
Hatti Mafuezi truca$r a9 -1lr beraber gevrsnin dl €lementini a!'rraq'
Bu haldadl=adp.
Diametrden olan it masafosi
r.=suptn)kimi baSa dlfuhlr. {rp }-noqtalar goxluludur. Sonsuz logik deqiqlikle
,' =ro(c)=oainq, 90<9< +dg' o<90 <r'
Elementar ir nitoti"in statxtft vo etalgt momentlen uyEun olamq
agalrdrkr kimi olar:1
dM D = a' snd7, dl D = u' sin" CqUmumi ssaslarla miisyyoo mtEqrall tatbiq etselq allnq:
M D = oz li^dg = - ,2 as gl" = 2a2.-
o
t o = o'i "n' d p = lj r"^' ro r - *"c0
' o3 I r - \l' a3r* I lr - *' zoV o = il p - ;"i" 2e
)1, = T
15E5. ay =2qa- rz (a > o) va r = O ayrileri rls hiidudlenen prarabokr
seqmentinin Ox va @ oxlaflrD ltaz€rrn etabt mometrtini tapln.
H alll Hesablzma dlisturlanrdatr,6 ^ b "r,=ll/Glft'La,. ty= !,"VG\e' 3;' -
a
isti6d. edak. Buna gdrs awelca sorhedleri tapaq. Odr h,qy=2dt-x'
vg
arliklerini birlikde hal.l ederelqt=O,
alanq. Odur ki,
Y=O
x=2a (osr<2a)
3t7
' " = i'{l* - *)'^ = i?["' - "*.' i' *Y :
=llur-'.: C.{-: 4ll* =!rr,o -1l.ot *31 5 a o, t "3 )ln l("-" 5 " '
* 54,-4 -l2E ot\-a4 .32 -32a47 ) I3s lo5'
,,, =f ,, ,* ='f *r( ,. -,')* ='f(rJ - 'n l* =' o o ( o) o( ")-l,t ,t)l' rca4 32os Eo4-1.7-tJ[ = 2 - s,= s
15t6 Yanmoxlan a vo D olan birciu elliptik t6vbnin bag oxlannnezeren etalot momenthi tapn. G = t)
Ealll Elhpsn bag odan ele kocrdinant oxlandu. E[ip6in tonliyiniparamctrik gekildo yazaq:
{t = azin t
t; =;;; (o s r <zr)
r -nm 0dan f -f ,"tr^r ila r o{an a-ya kimi artu. Ellips sy.isininkoonlinant o:daona g6rs simmefiHiyiai nczors alsaq, hesablarnam birincirubde apar0 neticcni 4< vumraq elviriplidir. Odur k{yazanq:
tt
,, = 11. t; r,lu,l = | a3 1 ^a a, = \,t3 .'o J o
Lr, | 1t * * z,f a, = ! oF | (r, z"* u * *"2 x), =
x
= !,r'1 (). z " z, . )"."
*)a, = la3 * (1, *., 21 * 1,,"
4E = +
," ,.t,=+I x'Q){1fu\t)= 4I o'sin'tbcftsuc€sat =-00
tt
= +i t? "a2
t*"2 d = iof "-z za, .
= )S o,lq - "*
*Y, = ;, {, -,r" "): = *1587. r=ooosq, y=osrs, (91<asr) dairavi qOvsirr altlq
msrkazinin kmrdinadannt raptn.
/IaIIi Qovs or oxura nezoron srmmetrik d<fuEundan onun aSrhqme:l/xezi c@ 1) oun uzarinde yerlegir.
O:rrinin uzunluEunu ra onua stdik mome ini o/ oxuna nezsan
hesablayaq:dq
L=a ldg=l"atdl= 2aa '
-q 0
^q .d .> td ,M , = at
_lc,s @p = 2a" lcs dp = 2q' s;aflO = Zq' 3;^o .
Af,rhq merkazi or oxu iizerindc oldulundan
4=o'Odur ki, arcaq 6 koordina!fir tspaq:
,-\ -lo2dna -oiaa ^,- ' -/asina -\o i-, L (6,?' =L
[ -,u].15t& u=yz; ay=rz, (a >o) parabolahn ilo ahata olunrnus
oblasln a$rhq merkszinin koordinartlanu tapm.Halll Mnslord fiqurm noqtelori birinci koordinant buca$rnm
tonbolenino nezoren simmetrikdir. Odur ki, a[rltq morkezi tanb<ilenin
iizeriDd.dir.?ly=-, y =4x
tenliklarini birlikdo hell edcrek, seiheddi tapaq:
*-^=o + 14 -a3r=o = *(r'-r')=o a a=Q, v=e, 03rso.a'
Odwki,
,,, =i I a - lV =if o'i-,.,1,. ='6l. o) 6[ , )
=l*,i-r-,oli' =zot -yot -to3
f 5 4a J[ r 4 20
, =?["a-4'jo =l*.1- -r,,1" -r -, .., ^, -4"=l['*-;)e=lt.' -r'-.J1,= a- -o-=- '
BeleIkla,
" M, 3o3 3 9a
20 a2 20
AtrrLq mer*ezi onbolsn iizorinda oldulundan
, =9o'2Colmaldr.
22lSEg, \*l-rt, (0<rsa, osy<b) oblastrnur alrrlrq
a" b'morlcznin korortdinantlam tapm.
IIalli Ellipsin d6rde bir hissasinin parametrik anliyini )azaq:
{x=asinr' o<t<L.['=Dcost 2
Onda molum diisturlann kdmsyi ile alanq:
tna
u, = !1 r, ra*4,.) =!j * *"2,.,"*,a, = ! a2f "# a, =^ 2'o' ' " 26 2 i)
=!,*1(r-,a2,)at,,"4= j"r2(,i",-];,-,)V=i*r(,-r=*;
3q)
tt
M ,, = '1,(t')yk$Q) = ta si r . bcasr ' a cx tdt ='oo
.E
= o2 ul "*2,"aat=-o2tf ,*2,a1*"r1= -.2t lrr^l ,l,i =]!.oo"3lo1
BelsIkle,s=4(Htipsinsahasi)
4
oldulundan
, MY 4o M- 4h{=i= o' '=i=;'t5n, a=r(r-sint) v=o(t-.r"r) (o<t<zx) sikloidinin l-ci
arlost ve or oxu ila ohate olunmug oblasbn a[lrhq markozinin
koordinarthfln I taPm.trIalli Molumdur ki,bu halda
u - = L'i u, 6 = lot'f (t - *",f a, = 4o32i "io6
rdt =l: = y o s y < zl=,',,-2tot*-2 -- -; 2 l2 I
= Bo3'! "ino
ydy = o"l (t - *" z uf ay = 13 J( - : "o,
z7 + r "o"2
z y - **3 z v\y --000
=,'i(r- r.*zy ' )*l*"+)at -.' ,i*'r* ="'i(l-**zt.)*or)+-
-" Iit-*"'!6t")==,'(1,-1" ',*i'tr), -]"f i",-f'" ")l:"=+',, ='i* = "'i(
-,i,,r[r - .ou rf ar =
= sl' i Q - *,,Y a1-,,1.'f ,(] - z "*,,, l, r,),)= ""'s = :a2 oldugunCan c(6,4) agtllq m{kazimn koordinantlan
M! M- s(= i=*. 4=i=e'kimi b!,ln olunur,
eahEmalatMonentbr va afitrhq ma*azl
1591. Ouraca$ a ve hlitrdurlifyii _/r olan diiLzbucaqhDn dz otuacagma
nozerEn stiatik momertini hesablayrn.
g^tub, ?t.15gZ Kdedsri a-ya beraber olan dilzbucaqh beraberfaah tglucagm2ri,
tareflerindan har birna razeren statik momeotini hisablayrn
carrat: d- i.o'JZtss3. Obraca$ a vc hundii,rhyii l, otan simmetrik ;;t'd;*'".
agrltq mar*ezini taprn.Carab:A[rhq merk zi scqmentin snnmetriya oru iizarinde
7oturacaqdan ] fi mssafasindodir.
.51591. r---;y=\lr' -x' yanmgernasinin a$rlq merkonnin koordinatlanmEprn.
Cavab: ( =O,a =?!1sg5' Absis oxu vo ,=f,rr-? yaflmgevresi ita monaud ot
yanndainnin a$rhq merksziniu koordinatlann taprn.
Cawb:6=o;a=!1596 a morkozi buca$rm geren .R radiuslu gevra qdvsiiLtriin agrhq
markszinin taprn.**6' 4grrha morkszi q6wii. gsran merkezi bucafur tsnb6leni
sut _iizerinde nurkczloll 21 --2 mesafesindadir.
t|g7. Koodina oxhn ve ji * ,!y = Ji L.* , ito mehdrd otan fiquruna{rrhq markszinin kmrdinatlanm taprn.
1598Koordinal odan
yerlagm\ qrivsfr
,2 vzu or* ,rt
=l elliPsinin
ib msMud olan fiqunur
392
Cavat: a=i;+=i
birinci kradranrda
alrrhq markazinin
4a 4bCa\ab t=i.n--i
1599. -2 ,,2\*\=, ellipsinir birinci kvadrantda yerlason qovsiinib absis
a' b'ororna nezcon statik momentini tapm'
b2dCavtb: "n *?a,csina, bunda s -ellipsin ekssentrisitetidir'
n0. y=siar sirusoidinin q6vsii va absis orornun pargsr ils mahdud olan
irr = odan x: = r -ye qeder) fiqurun agulq mtrtazinin koordinarlann
EPm- cavab:1=!;'=l
160l' ,=J= ta y=vz xetleri ila mshdud olan fiqurun absis oxuna
l+ x'
koordiratlanm tapm.
nozorsn stdik momentini taPn.
absis oxu.na nazoran s6it monotniat tapm.
crv^bt;+l
1&)z ./ = sin.r va r = ] Ot ..+t."t wiio) xatleri ils mlhdurl olan fiqurun
crv.u' a*f .12E160i. y=r2 ya y="1; xatlrri ila meMud olan fiqumn absis oxuna nszaron
statit momentini taPm
CavaU: fi1604 y2 =*3 -*4 qapat xatti ile mehdud olan 6qunu alrhq me*ezinin
ft6.ldinerlurm tlpm' czvab: (= l''= o
1605. I x -r'\v=gle; +e o | 2sscir xeuini[' ,l.
)nfiteleri arasrnda Yerlegsnkoordrnatlan-u taPm.
393
abeisleri { ='a ve rz=a olfr
qovsun[n alftrq merkazinin
Cauab: t =o: n =ala *,4"2 -t
4,V'-lbirinci tagmm a[rl rq
1ffi.,=r(z-"inrlr=o(t-"o"r) tsikloidminmarkezinin koordinatlanru tapm.
Cavab: E, = n.4 =!q1fi7, fsrU. orail birinci tag ve absis oru ilo mohdud olan fiqunrn alrrhq
morkc,i nin koordinatlanm tapm.
Cavab: g = a1;4 =1o
l60E x -- aas3 t, y = asia3 r asroidinin birinci kbadrar[da yerlegm qovsiiniiur
alr\ merkszinin koordinatlaruu taprn.
Gvab: r= 2'-o..r=2,,
lilg. R radiuslu yanmkti,ranin a[uhq rnerkszi onun tmaasi ma*eszinOonsnoqadar nusdedadir?
Cavab: 1n.1610, y"615p613 s5rhirrin agghq merloziri hprn. 8
Cavab: A!u\ morkezi simmetriya oxu iirzerinde markozden 4mesafcindsdir. 2
1611. R radiuslu yanmdairenin 62 diametrino nazaren inersila momemlnitapm.
Gvab. T 41612. Otracagmm raiuzu R, hiindii,rluyu Il olan silindrin Va, setfrmi, fra?nio
silindrin oulra nezoen inersiya momeotini tapm.
Cavah: i,lN , buradaM-silintlrin ym ssthinin kirtlesidir.1613. .R radiuslu ffip ssthinin k0renin diametrina nezeran inersiya
momt'ntlni tapm.
cavab,lun2 .
2
Sr0. MtIoYYoN ixTEQner,rx ToQRtBi rmsABLAl\MAsI
f . Darbuuqlrlo, osltr. Ogar v = 71'; firnksiyasr [a'D] pargasmda
kesilmez vO diferensialla nan drrss Ve
h=b- o, rt = a + ih (i =o,l'.-,n') Yi = A'i)n
ondab
ilrw= tt<Yo * h * '. + Yn-r) + R,a
bu6d1. R, =9:yy'(o k. € < b).
f . Trqalat da*trr. Yuxandakr ilarsmelerlc yazanq:
1 NO* = o<b* + h + vz +. . + Y ,_t) + Rn
d
ar^a* n,=-(b-r?hz !'G) G.€'<b)'
f . po-lotih iaarr (Sitrqlsot dfr urul z = 2t gotursoh alarq:
l r6t* =lWo + vv,\ + 4Qt1 + v3 + "+ v2k-r) + 4Qt2+ v4 + "' + vv-2)l+ R'
tunaa, n,=-q:94 {c) G<€"<b)'
bU. l*aallarrra serheddini 10 borabar hissola bdlmcklo
2_diiabucaqllar iisslu 115 I = i {x& inteqralrru hesablayn'
I
fl"ltl B]uIaAa v=Ji. n=lo olduda l=f =0'1 olur' Bol80
n6qtolari fu=1, 11 =rg+}=!1, xz=1,2, 'xg =\g irteqralahr
tu i*y".- bdlgU noqtelorino uygun qiymetlsri yo = J'o =l'yr=,!4=tpt1, 12--1,@5, h=\140, 1'J4 =1,183, rs=1,225'
y 6 =1,265, Y = I,304, Ys = \142, Ye = \37 8'
lreqrallama dusturundrn istihds edarok:
I = 0,1(1,000+ 1,049 + 1,095 + ll40 + !lE3 +1,725 +1'265 +
+l'304 +\342+l) 7E) = 0'l'l l'981rv l'20'
Xotartl qiymatbrdirck. Bu
firnlsiyasrtn x=l olduqda en
lf'ktl<M,=1 nrrua-,2
halda tl,2l parqasrnda f'(xl = -l==2JibOfrk qiyneti 0,5 olur. Balolikla,
n, s$.r.] =o,ozs"22Belslikle, inteqralm qrymotr 1 r; 1,201 0,025 olur.
MUqayise eunok [gi,n I =? ,frat imeqmhnr Nyuron-I,eybnis
diisturundaa isiftda ederek hesabhylq:2_ I
I = l,f**=iQJl-0"\ztg.1J
Belelikls, toqiqi qiymot taprlan imeqralda yerlagir.I6ri hteqratlama serheddini iO barabar iirssayc bolrneklo trapeslor
1
iisulu ile I = I jrdr futoqraluu hesablayrn.I
/Talli frapeslsr dusnmrna osson
r = 0,t.1t1111* L04e +1,095 + 1,140 r-t,l13 +1,225 + 1,265 +
+ \3M + \342 + 1,37t) = 1,2 I t[!2] pueasrnda I"G)=- -L, f'trl=1. Bareliklo.
4r/xr 4
n, s$.r.1*qooz"12 4Nshaycq 1* l2lE t Q002,
1616 Simpson d0sturu ile t =l'll-?* inteqmtrnr 0,00t daqiqlikle0
hesablayrn.Ealll Ovwlca ele ft addrmr segmaliyik k! verilen deq(liyi alaq.
Onda
f@=Jl*l; f'{x)=-\^ll + xt
3
f{i=-J==-
f"(x't = ---:==-='./(l+ x")'
ttv 1'1=--L\-.{(l + x')'
lfttr! n sir^r oziinirn en t$5nrk qiymetini [0'l] parqasrn<h x = 0
ndqtosinda ahr: [r 1o1l = s o.."ri
*,,fio-olt-al=$., ,
Beleki, bu xcta 0,001{an kigik otmahdr' O'au $ < o,oor, yeni } = 0'5
(eger h =0,5 otarsa, oDda ,r4 : O,OOZS) gOt'nrmek otar' Ba59 s6zlc bir az
i irrt f.r*i"*.t laamdr. lpkin bu hesablama deqiqliyino heg bir zsror
""irrnir,
-<iii tore ki. Qiymalandinne zamiur mutloq xsta$n limiti
iltitrrilrr0{d,". -B"leliklo,
iazrm olan daqidiye gamaq 09iin inteqrdlama
tt trrd- yanYa billmek lazund'rt.
r = 0, 0,5 va 1 ugiir IG> = [+7 fimksilasrrun qiynatlarini
hesablayaq: /(00)=l'000, /(0,5)= 1,1180, IQ)=\4r42 OnagOre
I ^,
U.tloooo*+,1,1180+ t,4l12l = 1,1477J
Belolikla, aldrlrmrz neticem minde birlaro qedar yuwarlagdrsaq I ar l'l4talanq.-'iliiqayise iigiin verilen rnteqralr Nyrton l'eybnis &rsturu ila daqiq
qiymotini hesablaYaq: II r--; f -''
t = t'',h + x' & .-l l''[17* ] lo1' * Jr*'21 ll =612',lO= !r Jz + nrt + #)t - 1fl,1t+z * 0,8814) t !1478.
1 .'- -'- " 2'
Belalikls Simpson d0saru ile hesabbnm4 hamin inteqrallr qiymati t9
yorq dord doqiq rsqam vz!r.
1Cl
Qahqrlabt:
1617.D0zbucaqLlar diishnrnu tetbiq etnokle z = 12 olduqda
2frsA*
hteqratm qiymotni aqibi hesablayrn vo alnan naticcni d3q( haflsmiiqayise edin,C,a*ab-6,2E32.
Trapal* dlfiurunu tatbE drra*ta inteqrolhn i:esobtoym va xdan,qiymatlandirk:1618. t 6J:_ (z=8).
Ol+.r1619. t &J--. (n = 127.
ol+.xr1620, ,t
'rf-!---.i .,lt-';"i"2 ** (n=6).ol 4
^ Simpaon dhsturuau latbiq ama*b inteqralton hesoblaytn:t62L e
if* @=4). cavab: 17'333'
I1622 E __
I .13 + ersxdr (n = 6)0
1623. t? surx -I -tu
(, = l0).0.r1624. \ x&
in'. 'l (z = 6)'
Cavab: 0,59315.
Cavab: 0,83556 .
Cavab:1,4675.
Cavab:5,4024.
Carab: 1,37039.
Cavab: 0,2288.
39t
AIX)BtYYAT
1. f,M. Orn<reruornq 'K)"c ,E0Q€peuluarbuoro H ItHrer?a:Ibgom
rc.Ercaeuu", r.t. \ n, lo69'197 0'
2. YLVI. Iltrurc. AK' Eorpuyx, .x I' fafi' f'II' Iorogas
'Mgreugfl{qesxrd a[aJrrl3 B rpl {epax u gar&qot"' s' 1'
BBe,uemre B urEJE3, npoxspoEsr, Err€rpsJl 'Kwu 1974'
6.IL AeurqosF{ 'C6oplm( 3a'q6u u ynp!'orelsB 11o
MarExarf,q@xordy anarrry", Mocna, 1977'
P.F. Qahremanov "Riyazi analizdan m0hazirela'f" I hisse'
SrmqaYr! 2005'
3.
4.
399
MUNDoRIcATGidr ............I FOSIL. Andliza girr, ......,..................
$ l. Riyazi analizin asas mlayrylan$ 2. Ardrcrltqlar nezeriyy$i .......,...............$ 3. Funksiyanu limiti$ 4. F.nksiyrrrm kesilrnediyi . .................
n FOSIL. Birdzyganli funbiyanu d{erensial hesah .._.........$ l. Agkar fiu*siymn tt rrm3si ......,.......$ 2. Tars fimtsiyanuq paramstrik va qeyri-akryar gs&ilde verilnig
firnksiyalann toramasi ...,..-..................
$ 3. T0rsmmin hmdesi manasr ...............$ 4. Funksiymn diferensiah ...................$ 5. Y0tsel( brtibli t0reme ve dif€rensirl$ 6. Roll Laqranj, Kogi teoremleri...$ 7. Funksiyanm atfutesl ye erqlmaql .......
$ E. Qabarqhgm istiqmeti, eyiloa n0qtesi ...................................$ 9. Qeyri-mlteyyanliklarin agrlme"1 ......
$ 10. Teylor d{lstuu$ ll. Funksiyanm eksts€.mumu, Funksiyamn en b0y[k ve et kigit
qiymetleri
$ 12. Xarakteristik n@alere nezeran fimkriya $afikinin qurulmasrIII FOSIL. Qeyrl-nfra1yan inteqral ........... .... .... ....... ...........
$ l. Sade qeyri-m[.yyetr int€qrallar .......II/ FaSlL. Mielyan inteqral
$ l. Mlloyyan inteqral ccmin limiti kimi$ 2. Qeyri-m0ayyan inteqralm k0meyi ile mteyyan irteqrahn
hesrablanmesl
$ 3. fte qilmst teot€mi ...........,...............$ 4. Qeyri-nexsusi inteqnllar ...................$ 5. Sahalarin hesablanmasl ...............,..........,
$ 6. Q0vs0n uaurlugu ..............,...,.....r...
$ 7. tlocmlrrin hesablanmasl ...........................
$ 8. Fulsmcdan almaa sgtt sahahin h€ssblmmasr ........,............$ 9. Momertlerin lxsrblenme.rr. Alrlq msrkezinia
koordinmtlan .,...$ 10. Mlbyyen inteqratn taqribi hesablanmaql ...............,.........-......
Odebiyyu ...........
344t23461
7272
9t959tt0r114119120t24134
r3E141
1501502622A
2632E52923lE33E356378
3t5395399
Qapa imzalamb: 07.07 .2tN9
Qap versqi: 25Tiraj 200. Sifarig No 25
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