8.2 Integration by Parts

Summary of Common Integrals UsingIntegration by Parts

1. For integrals of the form

€

x n∫ eaxdx,

€

x n∫ sinaxdx,

€

x n∫ cosaxdx

Let u = xn and let dv = eax dx, sin ax dx, cos ax dx

2. For integrals of the form

€

x n∫ ln xdx,

€

x n∫ arcsinaxdx,

€

x n∫ arccosaxdx

Let u = lnx, arcsin ax, or arctan ax and let dv = xn dx

3. For integrals of the form

€

eax sinbxdx∫ ,

Let u = sin bx or cos bx and let dv = eax dx

€

eax cosbxdx∫ ,or

Integration by Parts

If u and v are functions of x and have continuous derivatives, then

∫ ∫−= duvuvdvu

Guidelines for Integration by Parts

1. Try letting dv be the most complicatedportion of the integrand that fits a basicintegration formula. Then u will be the remaining factor(s) of the integrand.

2. Try letting u be the portion of the integrand whose derivative is a simplerfunction than u. Then dv will be the remaining factor(s) of the integrand.

Evaluate

dxxex∫To apply integration by parts, we wantto write the integral in the form .∫ dvuThere are several ways to dothis.

( )( )dxex x∫ ( )( )xdxex∫ ( )( )∫ dxxex1 ( )( )dxxex∫

u dv u dv u dv u dv

Following our guidelines, we choose the first optionbecause the derivative of u = x is the simplest anddv = ex dx is the most complicated.

u = x

du = dx dv = ex dx

v = ex( )( )dxex x∫u dv

∫ ∫−= duvuvdvu

∫−= dxexe xx

Cexe xx +−=

∫ dxxx ln2Since x2 integrates easier thanln x, let u = ln x and dv = x2

u = ln x

dxx

du1

= dv = x2 dx

3

3xv =

∫ ∫−= duvuvdvu

dxx

xx

x 1

3ln

3

33

∫−=

dxx

xx

∫−=3

ln3

23

Cx

xx

+−=9

ln3

33

Repeated application of integration by parts

∫ dxxx sin2 u = x2

du = 2x dx dv = sin x dx

v = -cos x

∫+−= xdxxxx cos2cos2

∫ ∫−= duvuvdvuApply integration byparts again.

u = 2x du = 2 dx dv = cos x dx v = sin x

∫−+−= xdxxxxx sin2sin2cos2

Cxxxxx +++−= cos2sin2cos2

Repeated application of integration by parts

∫ xdxex sinNeither of these choices for u differentiate down to nothing,so we can let u = ex or sin x.

Let’s let u = sin x

du = cos x dx dv = ex dx

v = ex

xdxexexe xxx sincossin ∫−−

u = cos x

du = -sinx dx dv = ex dx

v = ex

∫− xdxexe xx cossin

∫= xdxex sin

xdxexexe xxx sin2cossin ∫=−

xdxeCxexe x

xx

sin2

cossin∫=+

−