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

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

Example 1:

polynomial factor u x

sinv x

LIPET

sin sin x x x dx

dxxx cos

xv cos

dxvuuvdxvu

1u dxvuuvdxvu

cxxxdxxx cossincos

Example 2:

logarithmic factor

LIPET

dxxln

1v

dxvuuvdxvu

xu 1

dxvuuvdxvu

cxxxdxx ln1ln

xu ln

xv

1ln x x x dxx

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

dxvuuvdxvu

u dv uv v du

The Integration by Parts formula can be written as:

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

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

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.

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:

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!

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

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