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160. 161. 162.
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:) =~1 [log ( + 1) -Iog }
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=.
1.
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v
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l)
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1
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4
2
8
2 2
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.
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=
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+
1
.
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.
100.
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,
.
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.
1
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1)
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t
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, (16),
.
.
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,
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.
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.
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22
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,
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.
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1
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,
18).
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-
:
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.
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:
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1
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00)
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,
It >
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11
dx 1- (1'lm 1 -x10gpx 1- 1I-+",,10gp -
1
1 )- -1- ~--1 -. n 10gp - 1 2 p-l10gp - 1 2
ICoepnapa
2',
II=
t"logdx
1
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"-+""
f3
xlogx,log10gx-2--"
+ .
11-+""
30 , l1 ~ ~
1
1I=n
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,
"
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tg
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,
,
llm
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114...2
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- 1
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1 2
1 3
1 2
157.
.
-
,
( ),
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1 . lJ , 1 . .u , 1 .1
u l1lr .
38
rII
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Jl
(48)
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> ,.>"+ > '"
(>).
(49)
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,
(48),
t1 .
(49)
(49)
(50)
S2,.=u -[(u,-U 2 )+(ll -U4 )+ +(Uln-a-U2n-2)+U2,,-IJ, ,
(49)
(50)
, S2,. , h , . SZn .
S2n
-+
(N2 151),(47),
.
S2n= S. I/-too
S2 .. + 1
.
= S2n + " 2n
-
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lim 2n =0,
I/-t
S2n+l = (S2n+U2n)=
n-t
n-t
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n-tac
pe~
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,
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l
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"
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_
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11
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111/.
lIQAa-
Bo..asa
Jl
x(a. ). rAe tluuJUJ NOII.ClJlOHIllll. llltJAa
!.
" ()
. u. lI. " [, ) (..
,n
a,,=a"+I+ Q "+I+ . +Q"+,,
(=I,2, .)
'.
,
I n +1()+"+I()+ ... +,,+I'()\ -In+V''+I()~
"':+QII+"vlI+,(x)l-la1 vlI +1 () + (~a~,JJ1) vlI+.(x)+ ...
+ (,,- "_I) "-+, ()' = +
=I [,,, +1 ()- '''+1 () + ,["+I()- VII .. ()] ++,-! [VII +"_1
()- vlI+P(x)+cr",,,+,(x)l.
72
",
,
Vn +P_l(X)-Vn+(),
(=2,3, .. )
ln +1 () +n+l () +
." + n + () 1< 10'11 (V n +1 () - Vn+l ()] + ". + + 1 1.
"
1 1 ~ >(;> ... >(;)"> ... [,
R]
; (!!
167),
n>N(),
IR"I=la n +1 Rn+l(;)n+l + "'I n
< ,
lt,.
1l, "
=,
(12)
11......(9).
. 10g (1 + "n) 1 l --""-'--~(10),
],
,
(10) , , ,.=I+,., (n=,
(9) 1(0, (12), (11) .)(7)
1,2, ... ).(1+n+)-IIN'(s)
>. Jl, "
l1POU3BOIl (9) 6 , . , . .
(9)
n
,
11=1
[1- (-I)II~]
)
aIlCOJl,lllHO
It,
11=1Atr8IIp11l.
[1- -
~
11=0
"" 11=0
(1 + 1 1),
.
(9).
ct>
KO'l. KOHLepreHTHor .
1.
Jf
.1)
-
h :
10,
=1
.. ( ' 1)1+~
( 1 - 11=2
1)
>1
-< 1,
"" 1 ~-,11=1
2.
1t
(N!! 131, 1)2
"2= :21 2n--l'21"-1 ='''5'" 2n-l' 2+l'"
2n
2
2
4
4
2
2n
1)
.=
""
(1 +PIfIJI)'
~ ll',II=
vI(, ~
141
1
n=t 4n 2 -1
l(. l )
30.
1 ~= 2n-l . 2n+ 1 =J.-.~.~.~ ... 2n.,.-1 . 2n+ --2n 2n 1 11=1 2
2 2 2 4 4
4 n2
-
11=1~ ,
4n 2
.
.
1= ( 1) 1--11=1 4 n2 ~ 1 . ~ -.- .
40..
11=1
f.!.. .- (1 + _1_) 211 2 211 - 211=2
1I=14n2
,
50.
II~ [1 + ( ~ )211] =2,
(1 +n=o
_1_)= 2li 2211+ 1 == 2li (22 + 1. 2' + 1 ... 2211 + 1)== 2211 22
2. 22112211
11-+ 11=1
184.(13)
.
(1 +1111 ()],'.
11=0,
1) or ~.h
.
1
11=1
1.il,.tClO
- -
4 2 -1
1
== lIm - - = - ,II~ CIO
n
4 -l
1 4
'. .
2) ~+4+'" +2n:=~(1+2+ ... +11) :=211(1+11)
142
" () [, , ,
, .
/()= l+ (),
...
/ () = .. ..
[1 + () [1 + 1 ()],..
..
..
. .
...
..
..
,..
1 .. (x)=(1+u.(x)][1+U 1 (X)] ... (I+un{x)],...
..
..
..
.
..
.
.
.
,
(13)Q,lt 8
10 (), 11 (),
. ,
'n
(), ....
(13)
[, ),
,
1()
.. ~
n=
~ " (), ~ n(),n=
.
(13)
-
(13)
-
...
,
>
(1) h
[, , p~ ~ " () n;;;:
Jl.
npu.Alepu. 1.
n=2
ii (1 ~)
ICOHeprapa
> 1,
!n '20.
>
1
< < 1,
.
n=1
[
1 + -'----!.-"
1,
~
-
.
' -
6flf#. .h :
18.
3.
v ....()=-
aPOi8OI8
143
( 14)
1 ._--'--(1 + :)~1+
11=1
.})8 .
. , lf
1+
=1+ (- 1) +0 ( -1 ) ,1) n400, 2n 2 .%(-l)
II~.
2n
,
n =- % 11=1
1
11
1+
=----~--~-------
(n+I)':n! (+l) (+2) . (+n)
;.=-
1 " ( 1+ 1+ ~ 11=1
~ ) ( + l)=-n-
:-(-+-}-){--+--2-)-. -(-+-n-)
roje
(t.5)
()
=
II~OO (+ l)(+2)
.
n! n
. .11) . .. {-I)
.
(16) -",
(+ I)=li ()
11-+
n =, (+ l)=xF(x). + + }
'.
Iteo
nOlfll8tl ,
F(m+l)=mF(m)') P888ll'hl (1 +"; .'-01 ...,.,.. .......,.
1+ : .
'. HP
"+I) uat ( ...... " -.-1
144 , h
= 1, 2, ... ,
h
(1)
= 1/)(14)
r(m+l)=ml.(16)
(16')(+1)=()=n=l
(1 + ~ ) 1+ ~
20.
iv-
(16")
s6+isi6=(s+isi6),
(i=V -1),;,h
,
sin = cos- 1 Gsin - (-l) (-2) cos-se sin'e+ 31
sin - = cosm- 1
(
- 1)3!
( -
2)
sin 6
cos - s s 2
.
+
, .
= 2 + 1,
liapHU cos2~e==(I-sin2e)k,
sin
(2+ I)G=sin (si 2 ), (
(17)
sin (2 + 1) 6 = sin n (), (sin 2 ) = ()
= sin2 ),sin2 6 .
-
1 , 2 ,
n n(u),
(17)
sin (2.+ 1) = (- 1 )( _ 2 )sl
'
...
(- n ), ( = sin2 ),
(18)
Si(2.+l)_(1_~)(1_~) ... (1-~)'sl 1 2 n
sin =F ,
(2+
(18)')~
,
l)&=k,;, (k= 1, 2, ... ,
l(
')
n n!n
(15).
v r " .
146
e =--. ~II-I-lt
2. '.---..... 8"---. 211+1 211+1 10..,
III Jl 4tCll ~pa trnI -. IIJII801ll. " (.)111
==
. , Vl -
S - ' - . ","'''' .,'" 't
:_!I
~
211+1
2. '.-. .... 2n+l
,. - BIl , . -
=-
.. :_!I ,.. -.----. 211+ 1
. I. 8l ...
(11) ..oua
l1 + 118 =211+ l-....
C1'Ol'l ...... .. (18), IC ...... {'"+ 1)'- ............. ,
.....,(19)
.-(211 + })lib_L-(IIi~) 211+ 1in,__
. . (1- sin.~).siR.-l!~
2+ 1 2 1 1 + I n ,....
......
Jl.
(19')
stn =- f)
vt- ,
r.u
(20)
(11)
..... h ~
......(11) _. 1(8 , ,....
"
(20)
(20),
" . .
' (2 11+ 1)' l' l 'ln---, UD
$1"4
'.
211 + 1, _ ..... sin ...l.!-
211+ 1 .. l; ('........ 12 , k),
,. 8-
2n+l
(21')1
Ut-1im11-+0.
Ur)- X(l-3!)(l-~) .. ,(1-~). 4.. .1.1..
".u ..
t46 (22) ,
v1n),
,
n )==(I- 4(/:1)2) ... (1 - 4;2) ,(21),
(23) ,
1> V.!rn) > V~) ,(21),1)2
S12 --.>), "'''
-.I
~ q
" ..._
.. ,.,..........
.
117
.... _>0
fI
~
sin ct "" ., _ OII'I_II . ~.x." ." ." '''+1XI
. ,
(8)Cor. .
(7) (1).
....
.. I~tll () . , ()
"
(9).
f
...
_
_, u
f(x)cp(x)dx
cpelOj t>
(.NI! 133),
h
1, uulIiu,
.
''Ijt () I ~ 1>
160
," ,., """81
30.
(9)
,() Jl' ,() == 1... (> ). KONNpr.,al4ll > .(9) Jl
(>
Jl
0)1)
" " f !(X)dX _[J(X)r+cfJ(X)dX
OAaKJle
+!
. ()
... .
f
!(X}dX. CI (J(;r)tlx.
.pu OII. > .
IJ J(;r)dxl~+1
J!J(X)/dX < MJ~.+!
:.) 8.
1'0
pu '(~" -
'
+ 1
CI
> .
IllII..lU - f(x)tlx ItO"pr",.
(JrptUlll'f'Ha "ull 88 . . . ;r
f
. . . ,
,() ~.....,
>0, . " 8111krfU (1) ". . .",......- _
IIIII
(10)
,(.) ,(.) orpa.....
II - . I')'JlIf Illlel'pU
"1 I ~reIUljIf .... - f(x)dx. 1.11.
i ~ f !(x)dx=J(&>:"J(x1 ), !(x)dx:xJ(x.)-(,),
f
... .-+- , S-+- .-+- '" ".II .ll (10) . .ll 1 -+- .. .-+-, 111
JllltpaJl (9) ll.
11
'
')
......
..,...1..... ,.. . .1I'I(%)lIx,
.....,....
. .) ,IIO ..,.,IOC...., . ,...... ()-
U=:II'
.= "
Jf(J()II....
S ,.... ........ NI ;?:
.
J~X~
IU.Alu.
lt1'er/ _.... r8
161
-
}. n
JcosaXdl!k 2 +X8
. ' ll
10.,
,
.
I:~+:./' k'~X" ::/:./ dx', Jk"~X".....
__ n lC.
. . trrerp.
,,-'"
dx
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.
1 + 8 < .
1 ">0, >).' I8epnrp8, .
2i"
> . %).-I-CIC:;::.
'
>
1..
50, lI1'ePaJI
If - d% 1< fI}. .-.& os 1 < I...- .
.
f%)..- os.8
.... .
dx. (>"
> ,
>0). ~. OPrttPa.dx.
60.
. II'fetpJ,tI x~I.
...... aClecI.8It
............
.iI
41':. (:>l). fCODepnIp8..
l
" ona.ajyba 31
> ,
ff()d, f) s1
dx'.
te, 811
064'
" /ltuJelUz.
lIII1'eI'Pan
. h ,
810 . 1 aepnt. h uepnt, 2n+ .lt.
-
2n+l
. r.lte
!
50. ~ '" _in " d% l'll .. - 1, 11
.
171'
.
s"
_in t -d%= -1 ~d,,(X!'=t).11
.
>, !'
.
t
"
11 ~U
.81.(19)
.1IIIIIJL
llll % (,
- q). 6-Il11
-
f ()
f(x)d%, (0 Q8uu AltUlOAl
If f (]) f )
~
dx ,
=f f () dx < I)6-11,
I
".
0< I}s < I}l < %0(8), (
-
< b-ql < b-I}I < }.
" -4).
11,-+0 11.-+0 -II.
f () dx =,')"
IJ1)
= -
1
!Jf(X)dx\ () 1.
--l
II () I dx < I () I dx
--l
l
10. (N!! 187).
20. q> () Mefba ~ 1) u
(26) n iiUl ll
Ul uclllOBpeMeHO .
Ul
f
'
1= ,
q> () dx
Ul
f1()
dx.
(26),
(l-e)q>(x)
18014
r.uea
. li . l' Ir ... l .' ..
l-.. l-
.
-11 . ----"----- = - - 1r"'I,f
.
1
1
,1++ 2 +
=-1 . .
1
',~
;Og dx,l-ll
= .ll , , x~
log . 1 x 4l Iogx - --::::::::::=-.-l-2 x41 Yl- 22
' (0 .
.141-1 (I-)-1:
1 (1- X)CI
=x a - 1 (I-) CII+-- 1 _1,
--l
+b-l=,
,
(28)
> , > .
.. 1- < 1,
.
> .
80.
(29)
() = ,-1 - dx
S
1
.1=0
< 1.2)
() = - 1 - dx + - 1 - dx.
> ,
-1-:_=+-l-_l, -- +-l-0;
1
1)8)
( l4).
. II II, .. .
11
.". eor...... IlIUI)
183
II1n , ,-l
e- 7t : - 1 ... sCI+ a - 1 - - , xtt
n
+ - 1.
(29)
> .
CI> 1.
90.
n
- - - dx... V dx + dx,
>
. . .
100.
n
(30)
log dx == log dx + log dx
1
1 +2
1 +2
1 +2
1
n .. .n ,
..
log 1 , log --:-= ,1+2 1+2
-
,
,>),
-l 1 +-l 1) _-:-=!---~1, x~O, 8 o:+a-l=, ::=-l-. 1+1( : l+
-1 1 +-l 2) _-:-=~.-~1, 8 x~oo, l :+-l=I, 0:=2->1, KOaeprwpa
(
Sf ) d:
...
> ).
J-l[-' (ax)dX -S~(~)d]c~8
.. ...
" '
- [ '() dy ...,
S 1() dY] - S -+
...
d
1() dy ,
"
Il
(,
128)
(31)C~O
-lim f(A>,
Sdy -lim. !(i) log ~ .. /(0) 101 .!., ( < 5< ). . . "
cos - cos d S
...
.. 1og-,
S cos
.....
d: (!! 187, limf(x)=f(oo),x~
30.).
"
) ) >
d
...
(O ,
I'-"l
< 8 ()"
JY'-Y"I
< (),
'. ' ", " 2 (). n ( , ) 2 (n) 8
< 1I (', 1') -
f(x", ") 1
< 1I (', ') - f(x o )1 + 1I ( , ) - 1(", y")I
I
u IUlre.
195
I(,I) (,), - ,
10.
.
10.
, lllt lt {(,I)
1
. ptIHOAepHO llpeltUJlNa .
o4Ao'clllll. 20. l f (. ) UJI HelJpe1tUJlH4 06llJtl . Ilm
JlOBOJf1HO , ll4 ClJ4tredu. D(u, ') ()
}11(1')
d
..(.)
S duSF(U,V) D(cp,i')dv= SdVSF(U,V) D(cp,t>dU.D (, ) D (, ')
l()
111 (r)
(21)
= si
= cos v
= cos ,
v=,
si ,
----'~-'-
D (, ) == cos D (,0. I li1u J 2 f(a.,) .
Jg
!(, ) 2
.
IIOI'Y
RR It3aJI ( kU).
1,. no .0'8)' lICu8 JIC.II08e. Ta.1f.8 3 .
! l1UN.8 llpeJtDJla &t ,. ..
f a:f d1J f(ll.y.t)dll~'I
bt
41.
'} ouc 111=9 .. -. ,=0. .=19
-:
r-=Yt
224 ,
! (, , z)
(23)
I~
. 06 .(1,
dzj dyI !(x,y,l)dx= } f(x,y,z)dxdydz.
~
(23)
,
'
. ,
. Jl
,=., == 1 , == 2 ,
z= '1' z= '2'
2 2 ,
'1
=01'
(,, ')
! 1 (,,:) { ! (, , z):&
1 f(x.y,z)dxdydz= JiJft(X,y.Z)dXdYdZ. } !(,,,)
.lt
Oxyz.
l)a
06Jsa Jl . Jl.8
V . "t, "2' , 1In , ...If / f (,1, , :)
vtI
" uu11l iU'l u'l llI
S= .
l: MjVi,i=1
tI
=
l: ;V[=I
AforyhUAf
(24)
== ! (, , z) dv,f(, , z) 06lIfI ,. sp
f
U.
ll.u
V
06
fllIlll4 llJpoclDpy" IIIt llvu
dvR ' .IIell2 .un ,. If
/# 1tQ
f (.1'", ')11
."ll"tluICQ 06Iu . RfiItrIQM-At '''.
8. Jlm
(24)
c.-r peJl.1IOC
~ 5i. f\j.&.) 11;0 .
i:"f
lim8 II'frpaa.
11-+'("'.1) ....;
.
a'Vl(~)
"
Q . ~ . ~. u;.
-
8 !
,.~ zdxdpdz.
],
:t4~n: .l' lUtrt 2 + i+::Z z. (32) h
=,2
228r
zdxdydz= dx18
V;''l-xfJ
f
Vr2-x2-y2dy
r
zdz=
~ dx
Vr 2-x2
f
(2 -2 _ 2) dy=
=1 -
r
3
! 4 (r 2 -x 2 )2dx=3
f2
:! (x=rcost), sin 4 tdt=-,z
4 16
, - 2 - 2 ,
V
2 - 2 - 2.ltJ.
h :
firf
dx dy dz = dx
f f f dy
1
2
3
dz = 6, =,
= 1, = ,
=2, z =, z =3, 1, 2, 3.
.
f f fdx
'
dy
xyzdz=
xdx ydy zdz=..:;.
f f fdx dy
x1l y 2Z dz =
f
dx
y2dyu
f
zdz =
lal1~
205.
.!)
f (, , z)
, Vj
Oz,
r = const., z = const.
Oz,
= COl1st.,
.
(.
h ~
h abcd ' ' ' d' = Vi = r 11r ~z, = , -;;d = r A~, ' = AZ142).
,
=r cos~, = r sin~, z = z,, n
1)
0
~OBY u.
I
8w n
229
ffff (. .,
z) dv
=
ffff (
cos " r sin '. z) r dr drp dz =rdrdrpdz,
= F(r,rp,z),
dv = r dr dq> dz
' ,
f fdx
2
+2-2
dy fz
V2 + 2 dz
,1f
f f fdrp rdr
"2
2
zdz
= rc: 2 z =
r
=1
= 1, = .
/
'.
.
142
f (, , z)
, Vj
= const.
-
230 ,
n44
, = st. Oz
,=
st.
Vi
Oz.
h
-.1.8, ad
=- ,
= si 9.1."
-
h
abcd ' ' ' d' =
=2 si .1. .1.9.1."=
....... ad
01 = sin 8
.
143
r
11"
' = .1..
,
= si 8 CDS , = si 9 si" z = cos . r t (, , z) dxdydz = ! ( si8
cos >, si 9 si" cosO) 2 si dpd,dO =(33)
i
f
h =
F (, ,. ) 2 si9dd:d918
dv = 2 si 9 dp de dcp iilm , ,
dx
r
"'-2
2 -2-2
dy
"-x-::-"2+y-e-+-Z-"2dz,
IlI
. 8er
231
(33),
. . ( ' ) ll +)l1I+2 11 ==,II.
.
-
h :
10.
I e-~-zldddz= ,Si8e-f>1d d&d,=1(
.2
!!2
r .
r
= dtp si 8d6 2 e-pldp= - ~ ,-'" + : -" dp,
. ( i ) r-
.
xll
+y
2
i-zl! =,2.
,-+
Je-pld = ~ , .
2
1(
,.,
2.
d'J si 6
d6Jp2 e-pI dp = ;
Yii.r
Iog(x 2+y2+z2)dxd)ldz= Jd~8", "
k
I
~
si6d6 IpllOg p1dp=
--9- (31og,-1), ( )
X2+y2+Z2"",2.:rr. ,.,
,.
xydx d)ldz == sitp cos, dcp Si' 6 d6 . dcp (1 + l! +yll+ ZI)1 .
(1 +2)1
.
2
1
r
'.
== "8 19 , - 81
1
1,
+ r2 -
1 ~ 12 (1 + ~I!)I! ''-
~.... ( ) 2
+2 + zl! = ,2.
.
.206.
Jl Jl
. '7""
232
.
r,lla,.
h q1,q2 qa/)
(34)
= il (ql' q2' qa), = 1'2 (q1' q2' Qa), z
=r:pa (q1 , q2' qa)
q1 =~1 (, , ')' q2 = ,; (, , ')' qa = -r:pa (, z). Vj I, V2 (),
, v(:> ~ 1 , ~ ll , ~ ", .~X"i'
~
V1")",, I =XH~XII;'"
"
n-+8 ~
f.!(51i.5I1i, ... 51ii) V(1)= .
... !(1 .2 ,
X,.)dV(II)..
= w
...
Jt(X1,XII ,o ,X,.)dx1 dXII .o.dX,.,
(511,511" . St) - ,..
V').
t (1 , ., , ,.)
,
- Jl. , n- . . -
It. t .
... ff (X1'~"""
,xn) dX1 d~ .. dx,.=
.
'n..
t< J
'1'.""',.
. D(qt,qll'" .,q,.)
) D ('1' ll" o"")ld(11 dqll" dq,.,
I
rAejeXj=epi(ql.q., ... ,q,.)(i=1,2, ... :n)aJ'1I . ./I ql' Qll'
q,.., w ,
208.
,1l L
-
,
II , .
! (, ),
. ! ~ , ! (, ), Jl l1lJl
.
t (, ) AnI). ~), 1:), ...
236 m ),
,
2
, h (m-1) h (1) (?) (m) 2 2, 2 ' , 2 ,
h ()
!~ 2 , . m ) = 2
.
m~OD
m )
h () , , .
, , ,
. , J~71) 2
(37)
()= f(x,y)dxdy12m )
.
(37)
~OD
-- ,
J~m), :::: (), .
= f(x,y)dxdy=~~ /(x,y)dxdy!. j~m)
(37)
1l0HBeprupa,
. .
f (, ) >
2 , .1)
(37)
2 , h f (, ) - f (, ).
I (, ) <
,
f (, )
2 ,
(37)
, ,
, .
(38)
"
JJlf(x,y)ldXdY ,~m)
,
(39) Q 2
{(x,y)=/1(x,Y)-/2(') ,)
I/(x,y)l=f1(X,Y)+f2(X,y),
11 (x,y)=f(x,y),
>0, f2(X,y)=-f(,) {(,)
1)1)
-' dx= -.!.. -; . . 2erpaJl
S
~
(40)
. .
+ =
l- -+ (1 +m 2 )-1
1
-l=-7t. -l'
1,
. ;
< 1 ,
1
h
. .
20.
< N < ~ (, ) ='f (, 6) < , > 1, < 1; ) 2 + 2 - m~
= .30.
112
' (,
2n
y)pxdy
(ll+ 9)
f
def 'it(p, ) 2-1
~~~~
~
40.
,
e-I-l dx dy = -+ -+
+
f-;
dx
+
-+,
f-
dy =:
J~)
~a
-~.
= ( " dx + H~ - dX) ( dy + - dY) = 4 '" -+-+
f
-
-+
f
-
-+
f
:
240
rna.a
50.
1 )
-fl~ 3
(1 + 2 + 2 + Z2)811 .~
dx dy dz
=
. si' (} si > cos > dp d(} d> -1t
fl~
(1 + 2)8
2
m8
=f
si > cos > d>
1" sin
.e d9 . 1im . dp~a>
(1 +2)8
=
~.l
2 )
-+>
flm)~a>
dxdydz =1im fdfdf
=~ 5
fdX f (1 +x+y~ d (1 ++ )=....! 1im f (1 +x~- d (1 +)=!.. 15
15
m
209.
.
-
,(, )
, 2 ,)
;..
f (, )
f (, )
( , ) 2 , 2 .. h 2 , 0 4),
(42)
( 2 )= f(x,y)dxdy.-.
2 - 2 2 =1= . 2 ,
(42)
2 -+ ,
J=1im2+ 0
( 2 ), .
= f (, ) dx dy =!~~o f f (, ) dx dy1.x 2 +y2+z2-m 2 =. .-.
f
1) J~) x=y=z=O 2) J~m)
= = %=0,
x=y=z=m')t(x,y)=1
, (,)=-- ==, 2+ 2
1
'-
=, .
-
')
Afo.
I .
8 r
241
(42)
iiu 1OHBepupa,
.
(42)
uu.
f(x.y)">O
2 - 2 , 2 ,
!"
(42)
.
2 - 2 , h ,(,) -, ). '(,)
, K(~), .. , (43) 1OHBepupa. u. 1 )
.
t (, )
( ) 2
. K(~), K(~) K(~) + K(~) + ... = K(~)
(44)
~p)-+O
2
fSt(x,Y)dXdY .lt-K~)
, . ~
K(~). K(~) , (44) 1OHBeppa. u.
(42).
/3
1) \ , .
f (. ,
,
(43)
s
16
8
111
242
r If (,) 1< -
(42).
,
2 - 2 , , = 2 + 2, > , ,
= cos 6,
= sin ,
If(x,y) Idxdy
m>), , 0 , '
> , .
+
,
1(, )
llli1 6 (8)"1
>
'
(16)
If t (, ) dx- f f(x, ) dx 1=' f '(, y)dxxt ~
"1
, 1""
> (8)
U [, d]. = " ll
' !(,) dx-
f (, )"
f
dx
1 =11 !(x,y)dx I< 8
(15), (16). . \, 20. I( If (, ) I < ,() l( > l( [,
d],li1
! (, )
ll l( I(R [, ]
(>) ()
I(u ll
(0,+00],
('=
f
f(x,y)dx
ll I( [, , , xt.
d]."."
(16),
.
> 1".> (8);;;;:' .
I !(x,y)dx 1:( f1f(x,y)ldX< f'(X)dX ( 2 , ) -+ ,
.
1 -+00 2 - , , . [, d].
(16),
(17)
40.
u f (, ) dx u [, d].
f
q> (. ) , .
> [, d]. 1)
1 1 > ) () > . >0 (20) , n (19) > > .l
20.
()= ( 2 +2)2
2_ 8
dx
,
) !(,) x[a,b-q]
= ,
yE[c.d)
-
(21)
! (. ) dx ()
xE[a,b-q] yE[c.d]. [. d]
(21) q -+ .
(21")
{} =: r ! (, ) dx =:
-
. ~
r ! (, ) dyd].
lD
' [,
(21)
.
, ()
8
' ! (, dx ! (, ) ,~ojy
[, ]
. 1. . (21') . () d. .. 6
6>
illllll8l (6)-t
Ic,
>
"-.
(22) lD
I ! (, ) dX'-i !(, ) dx 1=1 S f(x. ) dx 1
, ~ -
> . > , ,
_ :; t. ~ __ .
si t -,--d== -t-.dl:;::o(l) si
'
11
Jl I
259
t dt = !!... s dx = s t 2
(1)
30,
sin cos dx = ~ sin ( + 1) + sin ( -1 ) dx 2
1.
40. 2_ ll dX=[ [,( 2 +2)21}
1
XII +y2
].1_-1- l+ll
1],
2_2
( 2 +2)1I
dx=
q
q2+y2
_1")
1
-+ .
50.
(
< < 1)1
-)'
cos dx =
-)'
cos _.- dx +
-)!
cos dx
t
>
1
. ( = )1
COS -)'--d ),(1
'.
11
40.. . eAR8 .
f
f(x)dx . ,
III . h r,t -)! e-.l
.
1) llaeu saax ( 188) t -tdt
26060. (~
I
< < 1)1
cosxy dX=J cosxy dX+J cosxy dx
1
> t
> .I
1)1
I,
; dx
I~ 'CO~XY' dx ~ ::'
, ., ,
30.,
IJ COSXYdX/=/ sinXY;Sin /< :0%
1
, --+00.2)
-
1
70.
(> )
al+x
smxy dx
> %
I . '. I=SIDXY d
> , , , 30.,l-cos ~./ 2
.
' ' - --+ - 2 + 2
--+
.
214.
.
. ..- 10. f(x, ) ll > [, d).
.
(23)
() "'" f(x, ) dx
. ll (, d), ll [, d] .
1)
Jl I 81 .
I JI, .111 pyrpr --+ , , DOC.lle =
t,
.ll
~ COS I t J --dx=ya-l -dt--+oo, ta
l --+, =-.
'1
%
%1
11
ll
261
1-+ 10
() = '(, ) dx =' (. ) dx = ().1-+ 1.
f
uu l1i u [. ( 1
()=(N!! 195), (24)
f
> )
d].
%1
f(x,y)dx+
f
f(x.y)dx.
() = f(x.yo)dx+ f(x,yo)dx. ~
f%1
'
~
. " (%1 %1
< < 1 .
199).
30.
,
(30)
f(x,y)dx,
! (, ) dy:
"' , s.", . ... ,, > , > . u111
(31)
!~ dy f(x,y)dx= dy f(x,y)dx, J~ dxJf(x,Y)dY = :
d
,.
:
=, " 111
dx f(x,y)dy
111
t:
" '.
(32)
dy Jf(x,y)dX::: dx f(x,y)dy.
t:
(31).
(30)
, , ,
t:
d
dyJf(x,y)dX- JdX
!(x,y)dy, d>~.
d
8
~',
d - .ll. r
,
(32)').
1)
30,
n
(30)
.
264
40.
,> ulii
1 (, ) > " (, ') YE[c,d], u (23) [c,d]
J/'v (,) ~x
. u [,
d] .
(34)
'() = I'y(x,y)dx
[c,d].
(34),
(35)
() == " (, ) d
f
' ()= ' () [,
d].
.
20.
h
[' () dy==~
I
dy
jf'Y (,) dx == (,)- '(, c)]di
,
(23),
S'()d=(}-();
,
,
(35),
' () = I() == " (, ) dx.
JlJl Jl
(34).
(34)
, . ~
(35)
Jl
'() =SI',(x,y)dx= S !'.(,) dx+ f(x,y)dx+ .. + S f'(x,y)dx+ .. '
. -l
f
Jl
[; d). ,
(.N'!! 168,,
20.)
S'()d == S dy S f(x.y)dx= S dy [n~ ;
S fY(XtY)dX]= -l,
~
= 1I~
S dx Sfy(x.Y)dY==n~ S [f(x,y)-f(x,c)]dx= -l -l
11
265
=
f (: ) dx - ! (, ) dx=J () - ().
(34).
,
() = f (, ) dx ,
f (, )
[, ]., , {(,)
= ,
n
()=
f(x,y)dX=!
n -l
f(x,y)dx
(!! .
168).
10.
h
e-:- d
dX,I)(C>
,
d>O).
-" dx =
;,
( > ),
8
> > .
d J
dye- xy dx = dx
Jd - dy = - e- dx Jd dy d dx = =log -;- .
20.
n h
(36)')
()
=
f
'
'..-/r'1(
Stnxy -x-dx,
.
. (k> , >0).
13.
(Frulli- )
(.N!! 191, ).
2663
CeJU14 r
' ()= - Ic " si dx,
38
r
> > 6> ,>),
l1 .
' (>- -"" cosxydx=_k__ , l! + kl!
(k
() = arc tg 1... + .k
(36)k
= , = , SI
()=
- -x-dx
=arc tg
k'
(k> ,
> ).
k_O,(38)
f011
- dx == l arc tg - = - , .- k-+O k 2
SlnXY.1
7t
(\> ),
11 ' =
JSI:X~
.
.
_-
, llq
(38)
fIOCaje
.... stn
. 7t dx ... -Ilm tg - ... - - .
(39)
J
SlnXY
'0
- dx =
I
t-+o
k
2
~2" )' > ,
?, = , 0),
.. J'(y)=frlf-Sind.= -_
,s+kl
() .. ~ log (2
1 2
+ k 2 ) + .''.
(40),
()=. O-..!..lgkl+ 2
C=-J..logk2 2
() =
1 - cos
dx ..
1 ( ) . log 1 +. k2 : "2
,
_
268 n
8
(
30).
, == 1f
cos6,
= sin ,
I
dx
2 - 2 dy = ( 2 +2)2I
f
Z
cos 2 d6
dp .I
2 - 2 [ ] dx " dx >(2 +2)2 dy= - ' 2 +2 /= 1+ 2 ="4'I I I
I
n
, >
1
>
1.
40.
() == 1- - dx, ' () = - (1+) d~~-~~
xr
1+
( >0),
() =log (1
+ >
() :=: arc tg dx, (1 + 2)
dx (+)d:=:-"2'I
I
--
1
I
dy (+) d = -"2'1
-
I
,
~ ..
(+ ) dxI
-
11
n l
269
.
[0,1]
(
20.)
70.
Po;sson-
(41)
=
f
e-xl dx
=yt,
dx = ydt,
> ,
( 42)
= e-2 df.
f
e-yI
f-2 f
dy
e-y2tl dt
=
f - f ye-2
! dy
dt = 2.
,l)
2 = dt
""
f
- 2 (I+t2) dy = .2
f~ =~ , +t 4
1
2
=
f e
""
x2 dx=2'
v"1i
Po;sson- h h
:
I = -2 dx -' dy =
f
""
f
ff
OD
e-(x'+yt)
dx dy = d6
"" f f
l dp
= ; .[ 1) n
n ~ -02: = : ' = V 2 ,'.
...
dt -2 (1+12) dy,
11
( .), n
,,
y,-1'(I+t2) dt=,-yl.
.
t
. n
27080.(43)
r
()=
f>
e-X2
cosxydx, '()= -
f>
xe-x'sinxydx;
' () =.!. [- ! sin xy]~_-.!.. -: cos dx = _...1..- (),2 2 2
f
' ()
-:; --, () 2
.
()=
-1
4.
,
(43),
(0)5::' e-'d~V2;" ~=,
()
f =f>
-2
cos
dx
vit -=2 ...
2
111
ULR-
215.(1)
Euler- aJI .I
1
(, ) = x
O-
1
(1- )
dx,
, ().h
> , >
l- u11i .
(1).
10.
= 1- t
(, )= -
St
(1-1)0-1 tb- 1 dt=
fI
t b- 1 (1_/)-1 dl=B
(, ).
20. (1) h 1 )1 1
(,)-
S(l-x)b-1 d -;=
[ (I-)II-1]l
+-1
-l
(l- x)b-2dx 2 )
=
= -1 Sx O- 1(1-x)b- 2dx - -l ,- -' -'-l dx, (/
1
.
..
u=(l-)-l,
dv=xa-1dx=d-
&der-ollll .,
...
271.
.
8oJt8lUle
-l (. )=
8
-l (, -l)--8 (, )
.
(2)
8 (, ) =
. . 8 (, - 1), (> 1). a+b-l-l
. 1I0 . 8
l"I'Uy (1). _ It-II> , 1IOC'I'8}e ~ tJJty(3)
1ll8811 . pilna
(2)
01)
ul-
(11).
10.
(11')
'()= S
xa - 1
10gxe-x
dx=
S
1
xa - 1
10gxe-x
dx+ x a -1}ogxe- x dx,1
S
) >~ajopaHTa 1
>0
8 - 1 (-log ) dx
,
- dx,20.
..
- 1
,
< x ,log < -log ( )"
(27)
R(O):="J;Og () do= IITI0g () do- 1101 () do
,
R'(a)=log ( + 1) -log ()
(12),
R' ()= 10g
(28)1) alf=X. 1) ( 134,
R(o) = log tJ do:: (1og - 1)-h ;.saAnaK 330.).
111
ulr- n
281
(25), (26)
(27)
R (0)= log () da=Ro " C1 -10g 2n.
f1
,
(21) (28), Raabe- ~0+1
R(a)-f 10g()d==(lg-l)+lgV 2n , (>).
40.
Fresnel-
==(29)
f
cos
2 dx == sin 2 dx ==
;
V ;.
1 == ff -! cos 2 dx dy = fcos 2 dx f e-Y'dy == "2 X~
== cos ,
::::
sin
:rr
1 =
f fdO
2"
e-p2siu'e
cos (2 08 2 6) dp.,
i: 2~
cos2
6:: , pdp =
2 082
du
-
,
1 ==
f 2 cos 0 f edO2
2
utc'6
cos
du1) =~2
f 1 +tg tg 8 cos2
:rr
dO2
--
4
,
tg 6 == t,
(30)
1 == ~ 12 dt 2 ) = ~ 2:. '12.21+1'
f
2 4.
'.
1)
Se-utg'&cos
tg"e du= - - . l+tg'
) 3 n h n
'1 ==
dt 1'+1'
==
S tJdt
"+I
1 't=-.
282h
l1
r
(29)
() h Fsn/-
= cos 2 dx = ~
W'F ($) s s
.
50.
(1)
F ($)=
f
f(t)e-stdt,
s,
w
, Lap/ace- m.u
f (1).
(31)
> ,
f If (t) I dtJt.
f (t) = t",
,
(l)
F(S,n)_Soo tne-atdt=
_.. [tne- at ]"" +~J~,.-te-.tdt,$
($>0),
$
.
(32)
F(s,n)=
f
fne-stdt=; ... .$
Stn
1
e-. t dt=; ($,n-l).
-1,2
, h
(3)
F(s,n)=
2 1 1 nl . - . ---($>0). Stne-s1dt=- -n-l ... S :; S$
$n+1
s == 1
(2) ()
F(l, ) =(n+ 1) =n (n) "" 111'n'8 -
du t dt - dt --- S -- S ----8 - S14+1 - 14+1- "00"
+l-
111
ulr-8
283
, F (s) = _1_, s > k Laplace-s-k
f (t) "'" ek ' ,
F(s)=J e-ldl=_l_.s-k
k F (s) =- s2+k 2
Laplace-
f (1) == sin kt,
k F (s)= Je- s, sin kt dl= - - , (s > ).
s2+k2
