0-calc8-2
TRANSCRIPT
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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
=+