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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∫=+
−