23. Integration by Parts

7/28/2019 23. Integration by Parts

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Badlands, South DakotaBadlands, South Dakota


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Integration By Parts

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 Partsformula.


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u dv uv v du

The Integration by Parts formula is a product rule forintegration.

u differentiates tozero (usually).

dv is easy tointegrate.

Choose u in this order: LIPET

Logs, Inverse trig, Polynomial, Exponential, Trig


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Example 1:

cosx x dxpolynomial factor u x

du dx

cosdv x dx

sinv x

u dv uv v du LIPET

sin cosx x x C

u v v du

sin sinx x x dx


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Example 2:

lnx dxlogarithmic factor lnu x

1du dx

x

dv dx

v x

u dv uv v du LIPET

lnx x x C

1lnx x x dx

x

u v v du


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This is still a product, so we

need to use integration byparts again.

Example 3:2 xx e dx

u dv uv v du LIPET2

u x xdv e dx

2du x dx xv eu v v du

2 2x xx e e x dx 2

2

x x

x e xe dx u xxdv e dx

du dx xv e 2 2x x xx e xe e dx

2 2 2x x xx e xe e C


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Example 4:

cosxe x dx

LIPET

xu e sindv x dx

xdu e dx cosv x

u v v du sin sinx xe x x e dx

sin cos cosx x x

e x e x x e dx

xu e cosdv x dx

xdu e dx sinv x

sin cos cosx x xe x e x e x dx This is theexpression we

started with!

uv v du


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Example 5:

cosxe x dx

LIPET

u v v du

cosxe x dx 2 cos sin cosx x xe x dx e x e x

sin coscos2

x x

x e x e xe x dx C

sin sinx xe x x e dx

xu e sindv x dx

xdu e dx cosv x

xu e cosdv x dx

xdu e dx sinv x

sin cos cosx x xe x e x e x dx sin cos cos

x x x

e x e x x e dx


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cosxe x dx

u v v du

This is called solvingfor the unknownintegral.

It works when both

factors integrate anddifferentiate forever.

cosxe x dx 2 cos sin cosx x xe x dx e x e x

sin coscos2

x x

x e x e xe x dx C

sin sinx xe x x e dx

sin cos cosx x xe x e x e x dx sin cos cos

x x x

e x e x x e dx

Example 5:


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A Shortcut: Tabular Integration

Tabular integration works for integrals of the form:

f x g x dxwhere: Differentiates to

zero in severalsteps.

Integratesrepeatedly.


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2 xx e dx

& deriv.f x & integralsg x

2x

2x

2

0

xe

xe

xe

xe

2 xx e dx 2 x

x e 2 xxe 2 xe C

Compare this withthe same problemdone the other way:


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2 xx e dx

u dv uv v du LIPET2

u x xdv e dx

2du x dx xv eu v v du

2 2x xx e e x dx 2

2

x x

x e xe dx

u xxdv 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 todo with tabular integration!

Example 5:


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3 sinx x dx

3x

23x

6x

6

sin x

cosx

sin x

cosx

0

sin x3 cosx x 23 sinx x 6 cosx x 6sin x + C

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23. Integration by Parts