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8.2 Integration by Parts

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Summary of Common Integrals Using

Integration by Parts

1. For integrals of the form

xneaxdx,

xnsinaxdx,

xncosaxdx

Let u = xnand let d = eaxdx! sin ax dx! "os ax dx

2. For integrals of the form

xnlnxdx,

xnarcsinaxdx,

xnarccosaxdx

Let u = lnx! ar"sin ax! or ar"tan ax and let d = xn

dx

#. For integrals of the form

eax

sinbxdx,

Let u = sin bx or "os bx and let d = eaxdx

eaxcosbxdx,or

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Integration by Parts

If u and are fun"tions of x and hae

"ontinuous deriaties! then

= duvuvdvu

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$uidelines for Integration by Parts

1. %ry letting d be the most "om&li"ated

&ortion of the integrand that fits a basi"

integration formula. %hen u 'ill be the

remaining fa"tor(s) of the integrand.

2. %ry letting u be the &ortion of the

integrand 'hose deriatie is a sim&ler

fun"tion than u. %hen d 'ill be the

remaining fa"tor(s) of the integrand.

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*aluate

dxxex

%o a&&ly integration by &arts! 'e 'ant

to 'rite the integral in the form . %here are seeral 'ays to dothis.

( )( )dxex x

( )( )xdxex

( )( dxxex

1 ( )( )dxxex

u d u d u d u d

Follo'ing our guidelines! 'e "hoose the first o&tion

be"ause the deriatie of u = x is the sim&lest and

d = exdx is the most "om&li"ated.

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u = x

du = dx d = exdx

= ex( )( dxex xu d

= duvuvdvu

= dxexe xx

Cexe xx +=

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dxxx ln2Sin"e x2integrates easier than

ln x! let u = ln x and d = x2

u = ln x

dx

x

du 1= d = x2dx

3

3xv =

= duvuvdvu

dxx

xx

x 1

3ln

3

33

=

dxx

xx

= 3

3

23

Cx

xx

+=9

3

33

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+e&eated a&&li"ation of integration by &arts

dxxx sin2

u = x2

du = 2x dx d = sin x dx

= ,"os x

+= xdxxxx cos2cos2 = duvuvdvu

-&&ly integration by&arts again.

u = 2x du = 2 dx d = "os x dx = sin x

+= xdxxxxx sin2sin2cos2

Cxxxxx +++= 222

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+e&eated a&&li"ation of integration by &arts

xdxex

sin

either of these "hoi"es for u

differentiate do'n to nothing!so 'e "an let u = exor sin x.

Let/s let u = sin x

du = "os x dx d = ex dx

= ex

xdxexexe xxx sincossin

u = "os x

du = ,sinx dx d = ex dx

= ex xdxexe xx cossin

= xdxex sin

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xdxexexe xxx sin2cossin

=

xdxeCxexe x

xx

sin

2

cossin

=+