integration by parts
DESCRIPTION
IntegrationTRANSCRIPT
![Page 1: Integration by Parts](https://reader036.vdocuments.mx/reader036/viewer/2022082902/577cc7991a28aba711a16ec5/html5/thumbnails/1.jpg)
Integration By Parts
Start with the product rule:
d dv duuv u vdx 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
This is the Integration by Parts formula.
dxvuuvdxvu
![Page 2: Integration by Parts](https://reader036.vdocuments.mx/reader036/viewer/2022082902/577cc7991a28aba711a16ec5/html5/thumbnails/2.jpg)
The Integration by Parts formula is a “product rule” for integration.
u differentiates to zero (usually).
v’ is easy to integrate.
Choose u in this order: LIPET
Logs, Inverse trig, Polynomial, Exponential, Trig
dxvuuvdxvu
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Example 1:
polynomial factor u x
sinv x
LIPET
sin sin x x x dx
dxxx cos
xv cos
dxvuuvdxvu
1u dxvuuvdxvu
cxxxdxxx cossincos
![Page 4: Integration by Parts](https://reader036.vdocuments.mx/reader036/viewer/2022082902/577cc7991a28aba711a16ec5/html5/thumbnails/4.jpg)
Example 2:
logarithmic factor
LIPET
dxxln
1v
dxvuuvdxvu
xu 1
dxvuuvdxvu
cxxxdxx ln1ln
xu ln
xv
1ln x x x dxx
![Page 5: Integration by Parts](https://reader036.vdocuments.mx/reader036/viewer/2022082902/577cc7991a28aba711a16ec5/html5/thumbnails/5.jpg)
This is still a product, so we need to use integration by parts again.
Example 3:2 xx e dx
LIPET
2u xxv e
u xxv e
dxvuuvdxvu
xu 2
xev
1u
xev
dxvuuv
dxxeex xx 22
dxxeex xx 22
dxexeex xxx 22
cexeexdxex xxxx 2222
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dxvuuvdxvu
u dv uv v du
The Integration by Parts formula can be written as:
![Page 7: Integration by Parts](https://reader036.vdocuments.mx/reader036/viewer/2022082902/577cc7991a28aba711a16ec5/html5/thumbnails/7.jpg)
Example 4 continued …
cos xe x dx LIPET
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 8: Integration by Parts](https://reader036.vdocuments.mx/reader036/viewer/2022082902/577cc7991a28aba711a16ec5/html5/thumbnails/8.jpg)
Example 4 continued …
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 9: Integration by Parts](https://reader036.vdocuments.mx/reader036/viewer/2022082902/577cc7991a28aba711a16ec5/html5/thumbnails/9.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.
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2 xx e dx & deriv.f x & integralsg x
2x
2x
20
xexexexe
2 xx e dx 2 xx e 2 xxe 2 xe C
Compare this with the same problem done the other way:
![Page 11: Integration by Parts](https://reader036.vdocuments.mx/reader036/viewer/2022082902/577cc7991a28aba711a16ec5/html5/thumbnails/11.jpg)
This is still a product, so we need to use integration by parts again.
Example 3:2 xx e dx
LIPET
2u xxv e
u xxv e
dxvuuvdxvu
xu 2
xev
1u
xev
dxvuuv
dxxeex xx 22
dxxeex xx 22
dxexeex xxx 22
cexeexdxex xxxx 2222
This is easier and quicker to do with tabular integration!
![Page 12: Integration by Parts](https://reader036.vdocuments.mx/reader036/viewer/2022082902/577cc7991a28aba711a16ec5/html5/thumbnails/12.jpg)
3 sin x x dx3x23x
6x6
sin xcos x
sin xcos x
0
sin x
3 cosx x 2 3 sinx x 6 cosx x 6sin x + C
p
![Page 13: Integration by Parts](https://reader036.vdocuments.mx/reader036/viewer/2022082902/577cc7991a28aba711a16ec5/html5/thumbnails/13.jpg)
13
• In an integral, do only integration by parts in the parts that are not easy to integrate.
• If the integration by parts is getting out of hand, you may have selected the wrong u function.
• If you see your original integral in the integral part of the integration by parts, just combine the two like integrals and solve for the integral.
Integration by Parts: Tricks