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Inte= gration by Parts
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Inte= gration by Parts
There = are two techniques for integrations.
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0014.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Integration by substitution finds antideriva= tives of derivatives that are obtained by using chain <= /span>rule. 
There = are two techniques for integrations.
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0015.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Integration by substitution finds antideriva= tives of derivatives that are obtained by using chain <= /span>rule. 
Integration by parts finds antiderivatives of derivatives that are obtained by the product rule,
There = are two techniques for integrations.
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0016.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Integration by substitution finds antideriva= tives of derivatives that are obtained by using chain <= /span>rule. 
Integration by parts finds antiderivatives of derivatives that are obtained by the product rule, i.e.
There = are two techniques for integrations.
W= 47;
uv' += u'v dx =3D uv + c 
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Inte= gration by Parts
Integration by substitution finds antideriva= tives of derivatives that are obtained by using chain <= /span>rule. 
Integration by parts finds antiderivatives of derivatives that are obtained by the product rule, i.e.
There = are two techniques for integrations.
However, this version is not useful in practice because its difficult to match the integrand of <= /span>an integration problem to uv' + u'v. =
W= 47;
uv' += u'v dx =3D uv + c 
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Inte= gration by Parts
Integration by substitution finds antideriva= tives of derivatives that are obtained by using chain <= /span>rule. 
Integration by parts finds antiderivatives of derivatives that are obtained by the product rule, i.e.
There = are two techniques for integrations.
However, this version is not useful in practice because its difficult to match the integrand of <= /span>an integration problem to uv' + u'v. =
The following is the workable version.
W= 47;
uv' += u'v dx =3D uv + c 
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Inte= gration by Parts
Integ= ration by parts:
Let u= =3D u(x), v =3D v(x), then
W= 47;
uv'dx= =3D uv – 
W= 47;
vu'dx=
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0019.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Integ= ration by parts:
Let u= =3D u(x), v =3D v(x), then
W= 47;
uv'dx= =3D uv – 
W= 47;
vu'dx=
An simpler way to describe it is to use the notation of differentials:
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0022.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Integ= ration by parts:
Let u= =3D u(x), v =3D v(x), then
W= 47;
uv'dx= =3D uv – 
W= 47;
vu'dx=
where= dv =3D v'dx and du =3D u'dx.
An simpler way to describe it is to use the notation of differentials:
W= 47;
udv =3D uv – 
W= 47;
vdu
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Inte= gration by Parts
Integ= ration by parts:
Let u= =3D u(x), v =3D v(x), then
W= 47;
uv'dx= =3D uv – 
W= 47;
vu'dx=
where= dv =3D v'dx and du =3D u'dx.
An simpler way to describe it is to use the notation of differentials:
W= 47;
udv =3D uv – 
W= 47;
vdu
When employed, we matched the integrand to the left hand side of the formula, =
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0020_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1wAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAADX ACUAgAAAAAAAAAKyRI6py+0Po5y02ouzArz7D4biSJbmiabqyrZud7zyTNf2ja9xzvf+D5ztgsSi 8WgbIpfMplPpjEqnNyj1is2SrNqul8r9isfGMPmMxpnT7DZr7Y7LRfC5fV6/69n5vX/c9yeoFTho OFV4qMiUuOhY1Pgo6RM5aVkVcKnJmLnpCdn5KUoZOmqKeZqaVKra+sbqGntSKZtKW2t6iyuqu+vZ 66sJHGw5TCypkay8zNzs/JxRAAA7 ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0021.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Integ= ration by parts:
Let u= =3D u(x), v =3D v(x), then
W= 47;
uv'dx= =3D uv – 
W= 47;
vu'dx=
where= dv =3D v'dx and du =3D u'dx.
An simpler way to describe it is to use the notation of differentials:
W= 47;
udv =3D uv – 
W= 47;
vdu
When employed, we matched the integrand to the left hand side of the formula, then convert <= /span>it to the right hand side and hope that the integral on the right hand side will be easier.
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0021_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1wAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAADX ACUAgAAAAAAAAAKyRI6py+0Po5y02ouzArz7D4biSJbmiabqyrZud7zyTNf2ja9xzvf+D5ztgsSi 8WgbIpfMplPpjEqnNyj1is2SrNqul8r9isfGMPmMxpnT7DZr7Y7LRfC5fV6/69n5vX/c9yeoFTho OFV4qMiUuOhY1Pgo6RM5aVkVcKnJmLnpCdn5KUoZOmqKeZqaVKra+sbqGntSKZtKW2t6iyuqu+vZ 66sJHGw5TCypkay8zNzs/JxRAAA7 ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0004.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
xex dx
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0004_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhngAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACd ACUAgAAAAAAAAAKdRI6py+0Po5y02gSy3rz7D4biSJbmiWZHyrbuC8ffKtf2jbN0zve+vfsJh8RQ sIhM/o7KpjPGfEqnpSj1it1Ys9zptgtWfsPk4biM5p3T7Nq6DXe94/TTvI4X3fP8zr4PaBAQSFg1 WIgI8pdIt8gI5/jIFimJRllJdokJprnJ1emJBRpKNUoqZXrqlKoqdtjKxwpbdFFre4ubqwtRAAA7 ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0023.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
xex dx
Match the integrand xex to udv,
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0023_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhngAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACd ACUAgAAAAAAAAAKdRI6py+0Po5y02gSy3rz7D4biSJbmiWZHyrbuC8ffKtf2jbN0zve+vfsJh8RQ sIhM/o7KpjPGfEqnpSj1it1Ys9zptgtWfsPk4biM5p3T7Nq6DXe94/TTvI4X3fP8zr4PaBAQSFg1 WIgI8pdIt8gI5/jIFimJRllJdokJprnJ1emJBRpKNUoqZXrqlKoqdtjKxwpbdFFre4ubqwtRAAA7 ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0024.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
xex dx
Match the integrand xex to udv, specifically
let u =3D x    and     dv =3D e= xdx
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0024_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhngAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACd ACUAgAAAAAAAAAKdRI6py+0Po5y02gSy3rz7D4biSJbmiWZHyrbuC8ffKtf2jbN0zve+vfsJh8RQ sIhM/o7KpjPGfEqnpSj1it1Ys9zptgtWfsPk4biM5p3T7Nq6DXe94/TTvI4X3fP8zr4PaBAQSFg1 WIgI8pdIt8gI5/jIFimJRllJdokJprnJ1emJBRpKNUoqZXrqlKoqdtjKxwpbdFFre4ubqwtRAAA7 ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0025.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
xex dx
Match the integrand xex to udv, specifically
let u =3D x    and     dv =3D e= xdx
To convert, we need du =3D u'dx and v =3D ∫ dv
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0025_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhngAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACd ACUAgAAAAAAAAAKdRI6py+0Po5y02gSy3rz7D4biSJbmiWZHyrbuC8ffKtf2jbN0zve+vfsJh8RQ sIhM/o7KpjPGfEqnpSj1it1Ys9zptgtWfsPk4biM5p3T7Nq6DXe94/TTvI4X3fP8zr4PaBAQSFg1 WIgI8pdIt8gI5/jIFimJRllJdokJprnJ1emJBRpKNUoqZXrqlKoqdtjKxwpbdFFre4ubqwtRAAA7 ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0032.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
xex dx
Match the integrand xex to udv, specifically
let u =3D x    and     dv =3D e= xdx
To convert, we need du =3D u'dx and v =3D ∫ dv
differentiate
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0032_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhngAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACd ACUAgAAAAAAAAAKdRI6py+0Po5y02gSy3rz7D4biSJbmiWZHyrbuC8ffKtf2jbN0zve+vfsJh8RQ sIhM/o7KpjPGfEqnpSj1it1Ys9zptgtWfsPk4biM5p3T7Nq6DXe94/TTvI4X3fP8zr4PaBAQSFg1 WIgI8pdIt8gI5/jIFimJRllJdokJprnJ1emJBRpKNUoqZXrqlKoqdtjKxwpbdFFre4ubqwtRAAA7 ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0032_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCgAnAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQAF ACMAgAAAAAAAAAIdBGKGuZr3GmSx0uvwzHx7CVpdqJFZgKJHujZqVgAAOw== ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0026.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
xex dx
Match the integrand xex to udv, specifically
let u =3D x    and     dv =3D e= xdx
To convert, we need du =3D u'dx and v =3D ∫ dv
differentiate
du
dx
=3D 1  
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0026_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhngAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACd ACUAgAAAAAAAAAKdRI6py+0Po5y02gSy3rz7D4biSJbmiWZHyrbuC8ffKtf2jbN0zve+vfsJh8RQ sIhM/o7KpjPGfEqnpSj1it1Ys9zptgtWfsPk4biM5p3T7Nq6DXe94/TTvI4X3fP8zr4PaBAQSFg1 WIgI8pdIt8gI5/jIFimJRllJdokJprnJ1emJBRpKNUoqZXrqlKoqdtjKxwpbdFFre4ubqwtRAAA7 ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0026_image011.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCgAnAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQAF ACMAgAAAAAAAAAIdBGKGuZr3GmSx0uvwzHx7CVpdqJFZgKJHujZqVgAAOw== ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0026_image012.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQAa AAIAgAAAAAAAAAIGjI+py70FADs= ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0027.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
xex dx
Match the integrand xex to udv, specifically
let u =3D x    and     dv =3D e= xdx
To convert, we need du =3D u'dx and v =3D ∫ dv
differentiate
du
dx
=3D 1  
du =3D 1dx  
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0027_image013.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhngAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACd ACUAgAAAAAAAAAKdRI6py+0Po5y02gSy3rz7D4biSJbmiWZHyrbuC8ffKtf2jbN0zve+vfsJh8RQ sIhM/o7KpjPGfEqnpSj1it1Ys9zptgtWfsPk4biM5p3T7Nq6DXe94/TTvI4X3fP8zr4PaBAQSFg1 WIgI8pdIt8gI5/jIFimJRllJdokJprnJ1emJBRpKNUoqZXrqlKoqdtjKxwpbdFFre4ubqwtRAAA7 ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0027_image014.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCgAnAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQAF ACMAgAAAAAAAAAIdBGKGuZr3GmSx0uvwzHx7CVpdqJFZgKJHujZqVgAAOw== ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0027_image015.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQAa AAIAgAAAAAAAAAIGjI+py70FADs= ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0028.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
xex dx
Match the integrand xex to udv, specifically
let u =3D x    and     dv =3D e= xdx
To convert, we need du =3D u'dx and v =3D ∫ dv
differentiate
du
dx
=3D 1  
du =3D 1dx  
integrate
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Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
xex dx
Match the integrand xex to udv, specifically
let u =3D x    and     dv =3D e= xdx
To convert, we need du =3D u'dx and v =3D ∫ dv
differentiate
du
dx
=3D 1  
du =3D 1dx  
integrate
v =3D ∫ ex dx  
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Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
xex dx
Match the integrand xex to udv, specifically
let u =3D x    and     dv =3D e= xdx
To convert, we need du =3D u'dx and v =3D ∫ dv
differentiate
du
dx
=3D 1  
du =3D 1dx  
integrate
v =3D ∫ ex dx  
v =3D ex 
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Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
xex dx
Match the integrand xex to udv, specifically
let u =3D x    and     dv =3D e= xdx
To convert, we need du =3D u'dx an= d v =3D ∫ dv
differentiate
du
dx
=3D 1  
du =3D 1dx  
integrate
v =3D ∫ ex dx  
v =3D e 
Thus  exdx =3D
∫
 u dv   =3D uv –  
∫
v <= /span>du
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Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
xex dx
Match the integrand xex to udv, specifically
let u =3D x    and     dv =3D e= xdx
To convert, we need du =3D u'dx and v =3D ∫ dv
differentiate
du
dx
=3D 1  
du =3D 1dx  
integrate
v =3D ∫ ex dx  
v =3D e 
Thus  exdx =3D xe<= /span>x 
∫
 u dv   =3D uv –  
∫
v <= /span>du
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Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
xex dx
Match the integrand xex to udv, specifically
let u =3D x    and     dv =3D e= xdx
To convert, we need du =3D u'dx and v =3D ∫ dv
differentiate
du
dx
=3D 1  
du =3D 1dx  
integrate
v =3D ∫ ex dx  
v =3D e 
Thus  exdx =3D xe<= /span>x –&= nbsp; ∫ e<= /span>xdx
∫
 u dv   =3D uv –  
∫
v <= /span>du
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Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
xex dx
Match the integrand xex to udv, specifically
let u =3D x    and     dv =3D e= xdx
To convert, we need du =3D u'dx and v =3D ∫ dv
differentiate
du
dx
=3D 1  
du =3D 1dx  
integrate
v =3D ∫ ex dx  
v =3D e 
Thus  exdx =3D xe<= /span>x –&= nbsp; ∫ e<= /span>xdx =3D xex – ex ∫
 u dv   =3D uv –  
∫
v <= /span>du
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Inte= gration by Parts
Steps= for Integration by parts:
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0036.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Steps= for Integration by parts:
1. Match the integrand to the form "udv", select a factor of the integrand to be u and set the rest as dv.
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0037.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Steps= for Integration by parts:
1. Match the integrand to the form "udv", select a factor of the integrand to be u and set the rest as dv.
2. Take the derivative of u to find du.
  &nbs= p; Take the antiderivative of dv to find v.
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0038.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Steps= for Integration by parts:
1. Match the integrand to the form "udv", select a factor of the integrand to be u and set the rest as dv.
2. Take the derivative of u to find du.
  &nbs= p; Take the antiderivative of dv to find v.
W= 47;
udv = as uv ̵= 1; 
W= 47;
vdu <= /span>
3. Re= write the integral
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0039.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Steps= for Integration by parts:
1. Match the integrand to the form "udv", select a factor of the integrand to be u and set the rest as dv.
2. Take the derivative of u to find du.
  &nbs= p; Take the antiderivative of dv to find v.
W= 47;
udv = as uv ̵= 1; 
W= 47;
vdu <= /span>
3. Re= write the integral
4. Tr= y to find
W= 47;
vdu =
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0040.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Steps= for Integration by parts:
1. Match the integrand to the form "udv", select a factor of the integrand to be u and set the rest as dv.
2. Take the derivative of u to find du.
  &nbs= p; Take the antiderivative of dv to find v.
W= 47;
udv = as uv ̵= 1; 
W= 47;
vdu <= /span>
3. Re= write the integral
4. Tr= y to find
W= 47;
vdu =
Remark= s:
* Try integration by parts if substitution doesn’t apply
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0041.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Steps= for Integration by parts:
1. Match the integrand to the form "udv", select a factor of the integrand to be u and set the rest as dv.
2. Take the derivative of u to find du.
  &nbs= p; Take the antiderivative of dv to find v.
W= 47;
udv = as uv ̵= 1; 
W= 47;
vdu <= /span>
3. Re= write the integral
4. Tr= y to find
W= 47;
vdu =
Remark= s:
* Try integration by parts if substitution doesn’t apply
* dv m= ust be integrable so we may find v
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0007.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
Ln(x) dx
∫
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0007_image047.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhngAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACd ACUAgAAAAAAAAAKdRI6py+0Po5y02gSy3rz7D4biSJbmiWZHyrbuC8ffKtf2jbN0zve+vfsJh8RQ sIhM/o7KpjPGfEqnpSj1it1Ys9zptgtWfsPk4biM5p3T7Nq6DXe94/TTvI4X3fP8zr4PaBAQSFg1 WIgI8pdIt8gI5/jIFimJRllJdokJprnJ1emJBRpKNUoqZXrqlKoqdtjKxwpbdFFre4ubqwtRAAA7 ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0042.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
Ln(x) dx
Match the integrand Ln(x)dx to udv,
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0042_image048.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhngAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACd ACUAgAAAAAAAAAKdRI6py+0Po5y02gSy3rz7D4biSJbmiWZHyrbuC8ffKtf2jbN0zve+vfsJh8RQ sIhM/o7KpjPGfEqnpSj1it1Ys9zptgtWfsPk4biM5p3T7Nq6DXe94/TTvI4X3fP8zr4PaBAQSFg1 WIgI8pdIt8gI5/jIFimJRllJdokJprnJ1emJBRpKNUoqZXrqlKoqdtjKxwpbdFFre4ubqwtRAAA7 ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0043.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
Ln(x) dx
Match the integrand Ln(x)dx to udv,
let u =3D Ln(x)     dv =3D dx
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0043_image049.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhngAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACd ACUAgAAAAAAAAAKdRI6py+0Po5y02gSy3rz7D4biSJbmiWZHyrbuC8ffKtf2jbN0zve+vfsJh8RQ sIhM/o7KpjPGfEqnpSj1it1Ys9zptgtWfsPk4biM5p3T7Nq6DXe94/TTvI4X3fP8zr4PaBAQSFg1 WIgI8pdIt8gI5/jIFimJRllJdokJprnJ1emJBRpKNUoqZXrqlKoqdtjKxwpbdFFre4ubqwtRAAA7 ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0044.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
Ln(x) dx
Match the integrand Ln(x)dx to udv,
let u =3D Ln(x)     dv =3D dx
differentiate
du
dx
=3D  
1
x
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0044_image050.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhngAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACd ACUAgAAAAAAAAAKdRI6py+0Po5y02gSy3rz7D4biSJbmiWZHyrbuC8ffKtf2jbN0zve+vfsJh8RQ sIhM/o7KpjPGfEqnpSj1it1Ys9zptgtWfsPk4biM5p3T7Nq6DXe94/TTvI4X3fP8zr4PaBAQSFg1 WIgI8pdIt8gI5/jIFimJRllJdokJprnJ1emJBRpKNUoqZXrqlKoqdtjKxwpbdFFre4ubqwtRAAA7 ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0044_image051.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCgAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQAG ACMAgAAAAAAAAAIghGOJCavcoDuyshmvfbrzT3khmJHYuYXByjYs8iqrbBUAOx== ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0044_image052.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHQADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAb AAEAgAAAAAAAAAIEjI+pWgA7 ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0044_image053.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHQADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAb AAIAgAAAAAAAAAIGjI+py90FADs= ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0045.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
Ln(x) dx
Match the integrand Ln(x)dx to udv,
let u =3D Ln(x)     dv =3D dx
differentiate
du
dx
=3D  
du =3D  
1
x
x
dx
------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0045_image054.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhngAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACd ACUAgAAAAAAAAAKdRI6py+0Po5y02gSy3rz7D4biSJbmiWZHyrbuC8ffKtf2jbN0zve+vfsJh8RQ sIhM/o7KpjPGfEqnpSj1it1Ys9zptgtWfsPk4biM5p3T7Nq6DXe94/TTvI4X3fP8zr4PaBAQSFg1 WIgI8pdIt8gI5/jIFimJRllJdokJprnJ1emJBRpKNUoqZXrqlKoqdtjKxwpbdFFre4ubqwtRAAA7 ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0045_image055.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCgAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQAG ACMAgAAAAAAAAAIghGOJCavcoDuyshmvfbrzT3khmJHYuYXByjYs8iqrbBUAOx== ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0045_image056.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHQADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAb AAEAgAAAAAAAAAIEjI+pWgA7 ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0045_image057.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHQADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAb AAIAgAAAAAAAAAIGjI+py90FADs= ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0045_image058.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAAAa AAIAgAAAAAAAAAIGjI+py70FADs= ------=_NextPart_01C97CE5.613B96E0 Content-Location: file:///C:/8F3256F3/16IntegrationbyParts_files/slide0046.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
Ln(x) dx
Match the integrand Ln(x)dx to udv,
let u =3D Ln(x)     dv =3D dx
differentiate
du
dx
=3D  
du =3D  
integrate
v =3D ∫ dx  
1
x
x
dx
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Inte= gration by Parts
Example: Find
W= 47;
udv =3D uv – 
W= 47;
vdu
∫
Ln(x) dx
Match the integrand Ln(x)dx to udv,
let u =3D Ln(x)     dv =3D dx
differentiate
du
dx
=3D  
du =3D  
integrate
v =3D ∫ dx  
v =3D x
1
x