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8/8/2019 7 hng bin i cho mt bi tch phn www.MATHVN
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7 HNG BIN I CHO MT BI TON TNH TCH PHNNguyn Hu ThanhTrng THPT Thun Thnh s I
Mt trong nhng phng php hc ton l sau mi bi ton chng ta cn tm ra nhngim nhn c th hiu vn mt cch thng thi hn.
Vy lm c iu , ngi hc ton cn iu g?1. Suy ngh tht k, tht thu o v vn c t ra.2.
Tm mi lin h gia cc kin thc xung quanh vn .3. T t cu hi xung quanh mt vn nh tm cch tng qut thch hp.
Chc cc bn thnh cng ! !
Xin phn tch qua mt bi ton nh sau:
Bi ton : Tnh tch phn I =2
0
sin
sin cos
xdx
x x
Nhn xt 1: Quan st thy c hm s di du tch phn c dng phn thc. Vy kinthc s s dng cho hm phn thc l g? Chc chn chng ta ngh n nguyn hm
ln | |dx
x C
x
Vy s dng c cng thc ny chng ta cn phi tm mi cch bin i v dng !Nhn xt 2: y ch xut hin 2 hm s lng gic l sinx v cosx . Vy c cch no biu
din thng qua mt yu t khng ? Ta cng tm kim kin thc gii quyt.- Hng 1: Chia c t v mu cho cosx ta c
1 1 1( ) 1
cos 1 1 1
sinx tgx tgxf x
sinx x tgx tgx tgx
t t t= tgx
I =
dt
t
CBtdt
t
Adx
tt
tdtdx
2211)1)(1(
- Hng 2: t tan2
xt th
2
2 2
2 1- tsin ; cosx=
1 1
tx
t t
Vi hng trn ta c th tnh c tch phn c dng tng qut sau:
1 1 1
2 2 2
sin cos
sin cos
b
a
a x b x c I dx
a x b x c
Cc bn hy lm bi ton trn v t mnh ngh ra bi v gii nh!Nhn xt 3: Xut pht t quan h ca sinx v cosx . iu g c bit trong cn ca tch phn
?
- Hng 3: t x= 0 :
2 2: 0
2
dx dt
t x t
x t
( y l cch t n ph m khng lm thay i
cn ca tch phn c dp trao i cng cc bn).
Khi :2 2 2
0 0 0
sin s s
sin cos sin cos sin cos
x co t co x I dx dt dx
x x t t x x
2 2 2
0 0 0
sin s2
sin cos sin cos 2 4
x co x I dx dx dx I
x x x x
8/8/2019 7 hng bin i cho mt bi tch phn www.MATHVN
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Tht ng kinh ngc !!!!!
Vi hng trn ta c th tnh c tch phn tng qut sau:2
0
sin
sin s
n m
n nm m
xI
x co x
Cc bn hy lm bi ton trn v t mnh ngh ra bi v gii nh!Nhn xt 4:Nu dng bin i lng gic th nh th no ?
- Hng 4: Bin i f(x) =
4cot1
21
4sin2
44
sin
xg
x
x
v ta tnh c bnh
thng.- Hng 5: Bin i
f(x) =
1
2cos
12
2
1
2cos
2
2cos12sin
2
1
sincos
)sin(cossin22 x
xtgx
xx
xx
xxx
v ta tnh vi cc tch phn bnh thng ca hm lng gic.Nhn xt 6: V tch phn c dng hm phn thc nn nu ta bin i t thc tm cch
vit c qua mu s v o hm ca mu th hay qu !- Hng 6: Bin i
cos cos 1 cos( )
cos 2( cos ) 2 2( cos )
sinx sinx x sinx x x sinxf x
sinx x sinx x sinx x
Nhn xt 7: Quan st ticsh phn cn tm ta thy s sai khc ca t s v mu s, vy nu tatm c mt tch phn khc c h hng vi n th sao nh ? Tr li cu hi ta i xt tch phn
2
0
s
sin cos
co x J dx
x x
( gi l tch phn lin kt ca tch phn I) .
T hai tch phn trn ta i gii h :I J
I J
s tm c I .