6.3 Integration by Parts & Tabular Integration. Problem: Integrate Antiderivative is not obvious...
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Transcript of 6.3 Integration by Parts & Tabular Integration. Problem: Integrate Antiderivative is not obvious...
![Page 1: 6.3 Integration by Parts & Tabular Integration. Problem: Integrate Antiderivative is not obvious U-substitution does not work We must have another method.](https://reader035.fdocuments.in/reader035/viewer/2022062809/5697c01a1a28abf838ccec37/html5/thumbnails/1.jpg)
6.3Integration by Parts & Tabular Integration
![Page 2: 6.3 Integration by Parts & Tabular Integration. Problem: Integrate Antiderivative is not obvious U-substitution does not work We must have another method.](https://reader035.fdocuments.in/reader035/viewer/2022062809/5697c01a1a28abf838ccec37/html5/thumbnails/2.jpg)
Problem:
Integrate cos x x dxAntiderivative is not obvious
U-substitution does not work
We must have another method to at least try and find the antiderivative!!!
![Page 3: 6.3 Integration by Parts & Tabular Integration. Problem: Integrate Antiderivative is not obvious U-substitution does not work We must have another method.](https://reader035.fdocuments.in/reader035/viewer/2022062809/5697c01a1a28abf838ccec37/html5/thumbnails/3.jpg)
By Parts formula:
Start with the product rule:
d dv duuv u v
dx dx dx
d uv u dv v du
d uv v du u dv
u dv d uv v du
u dv d uv v du
u dv d uv v du
u dv uv v du This is the Integration by Parts formula.
![Page 4: 6.3 Integration by Parts & Tabular Integration. Problem: Integrate Antiderivative is not obvious U-substitution does not work We must have another method.](https://reader035.fdocuments.in/reader035/viewer/2022062809/5697c01a1a28abf838ccec37/html5/thumbnails/4.jpg)
u dv uv v du
The Integration by Parts formula is a “product rule” for integration.
u differentiates to zero (usually).
dv is easy to integrate.
Choose u in this order: LIPET
Logs, Inverse trig, Polynomial, Exponential, Trig
![Page 5: 6.3 Integration by Parts & Tabular Integration. Problem: Integrate Antiderivative is not obvious U-substitution does not work We must have another method.](https://reader035.fdocuments.in/reader035/viewer/2022062809/5697c01a1a28abf838ccec37/html5/thumbnails/5.jpg)
Example 1:
cos x x dxpolynomial factor u x
du dx
cos dv x dx
sinv x
u dv uv v du LIPET
sin cosx x x C
u v v du
sin sin x x x dx
![Page 6: 6.3 Integration by Parts & Tabular Integration. Problem: Integrate Antiderivative is not obvious U-substitution does not work We must have another method.](https://reader035.fdocuments.in/reader035/viewer/2022062809/5697c01a1a28abf838ccec37/html5/thumbnails/6.jpg)
Example 2:
ln x dxlogarithmic factor lnu x
1du dx
x
dv dx
v x
u dv uv v du LIPET
lnx x x C
1ln x x x dx
x
u v v du
![Page 7: 6.3 Integration by Parts & Tabular Integration. Problem: Integrate Antiderivative is not obvious U-substitution does not work We must have another method.](https://reader035.fdocuments.in/reader035/viewer/2022062809/5697c01a1a28abf838ccec37/html5/thumbnails/7.jpg)
This is still a product, so we need to use integration by parts again.
Example 3:2 xx e dx
u dv uv v du LIPET
2u x xdv e dx
2 du x dx xv e u v v du
2 2 x xx e e x dx 2 2 x xx e xe dx u x xdv e dx
du dx xv e 2 2x x xx e xe e dx
2 2 2x x xx e xe e C
![Page 8: 6.3 Integration by Parts & Tabular Integration. Problem: Integrate Antiderivative is not obvious U-substitution does not work We must have another method.](https://reader035.fdocuments.in/reader035/viewer/2022062809/5697c01a1a28abf838ccec37/html5/thumbnails/8.jpg)
Example 4:
cos xe x dxLIPET
xu e sin dv x dx xdu e dx cosv x
u v v du sin sinx xe x x e dx
sin cos cos x x xe x e x x e dx
xu e cos dv x dx xdu e dx sinv x
sin cos cos x x xe x e x e x dx This is the expression we started with!
uv v du
![Page 9: 6.3 Integration by Parts & Tabular Integration. Problem: Integrate Antiderivative is not obvious U-substitution does not work We must have another method.](https://reader035.fdocuments.in/reader035/viewer/2022062809/5697c01a1a28abf838ccec37/html5/thumbnails/9.jpg)
Example 5:
cos xe x dxLIPET
u v v du
cos xe x dx 2 cos sin cosx x xe x dx e x e x
sin coscos
2
x xx e x e xe x dx C
sin sinx xe x x e dx xu e sin dv x dx
xdu e dx cosv x
xu e cos dv x dx xdu e dx sinv x
sin cos cos x x xe x e x e x dx
sin cos cos x x xe x e x x e dx
![Page 10: 6.3 Integration by Parts & Tabular Integration. Problem: Integrate Antiderivative is not obvious U-substitution does not work We must have another method.](https://reader035.fdocuments.in/reader035/viewer/2022062809/5697c01a1a28abf838ccec37/html5/thumbnails/10.jpg)
Example 5 (cont.):
cos xe x dx u v v du
This is called “solving for the unknown integral.”
It works when both factors integrate and differentiate forever.
cos xe x dx 2 cos sin cosx x xe x dx e x e x
sin coscos
2
x xx e x e xe x dx C
sin sinx xe x x e dx
sin cos cos x x xe x e x e x dx
sin cos cos x x xe x e x x e dx
![Page 11: 6.3 Integration by Parts & Tabular Integration. Problem: Integrate Antiderivative is not obvious U-substitution does not work We must have another method.](https://reader035.fdocuments.in/reader035/viewer/2022062809/5697c01a1a28abf838ccec37/html5/thumbnails/11.jpg)
A Shortcut: Tabular Integration
Tabular integration works for integrals of the form:
f x g x dx
where: Differentiates to zero in several steps.
Integrates repeatedly.
![Page 12: 6.3 Integration by Parts & Tabular Integration. Problem: Integrate Antiderivative is not obvious U-substitution does not work We must have another method.](https://reader035.fdocuments.in/reader035/viewer/2022062809/5697c01a1a28abf838ccec37/html5/thumbnails/12.jpg)
2 xx e dx & deriv.f x & integralsg x
2x
2x
2
0
xexexexe
2 xx e dx 2 xx e 2 xxe 2 xe C
Compare this with the same problem done the other way:
![Page 13: 6.3 Integration by Parts & Tabular Integration. Problem: Integrate Antiderivative is not obvious U-substitution does not work We must have another method.](https://reader035.fdocuments.in/reader035/viewer/2022062809/5697c01a1a28abf838ccec37/html5/thumbnails/13.jpg)
Example 6:2 xx e dx
u dv uv v du LIPET
2u x xdv e dx
2 du x dx xv e u v v du
2 2 x xx e e x dx 2 2 x xx e xe dx u x xdv e dx
du dx xv e 2 2x x xx e xe e dx
2 2 2x x xx e xe e C This is easier and quicker to do with tabular integration!
![Page 14: 6.3 Integration by Parts & Tabular Integration. Problem: Integrate Antiderivative is not obvious U-substitution does not work We must have another method.](https://reader035.fdocuments.in/reader035/viewer/2022062809/5697c01a1a28abf838ccec37/html5/thumbnails/14.jpg)
3 sin x x dx3x
23x
6x
6
sin x
cos x
sin xcos x
0
sin x
3 cosx x 2 3 sinx x 6 cosx x 6sin x + C