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Form= ulas From Integration by Parts
------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0010.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Form= ulas From Integration by Parts
Integ= ration by parts:
Let u= =3D u(x), v =3D v(x), then
where= dv =3D v'dx and du =3D u'dx.
W= 47;
udv =3D uv – 
W= 47;
vdu
------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0010_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1wAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQDX ACQAgAAAAAAAAAKujI+py+0Po5y02osV2Lz7D4biSJbmiabqyrac4cbyTNf2rcL4zvf+L9MBh8Si sSY8KpfMZrIJjUptz6n1ih1Vs9zudOsNi4vgsfl8K6PX7JW6DY+H3vK6nG7Pr/H6vpjvF5gFKFgo RWiYuISo2EjE6BjZAylZSRVgmbmIqdn5yOkZOgkqWnppiopEmsrqttoKa0IZizpLW2p7G5qr28nb m/kLXJlRbHyMnKy8jFEAADs= ------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0011.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Form= ulas From Integration by Parts
Integ= ration by parts:
Let u= =3D u(x), v =3D v(x), then
where= dv =3D v'dx and du =3D u'dx.
W= 47;
udv =3D uv – 
W= 47;
vdu
The integrals of some functions require using the integration by parts many times. 
------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0011_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1wAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQDX ACQAgAAAAAAAAAKujI+py+0Po5y02osV2Lz7D4biSJbmiabqyrac4cbyTNf2rcL4zvf+L9MBh8Si sSY8KpfMZrIJjUptz6n1ih1Vs9zudOsNi4vgsfl8K6PX7JW6DY+H3vK6nG7Pr/H6vpjvF5gFKFgo RWiYuISo2EjE6BjZAylZSRVgmbmIqdn5yOkZOgkqWnppiopEmsrqttoKa0IZizpLW2p7G5qr28nb m/kLXJlRbHyMnKy8jFEAADs= ------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0012.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Form= ulas From Integration by Parts
Integ= ration by parts:
Let u= =3D u(x), v =3D v(x), then
where= dv =3D v'dx and du =3D u'dx.
W= 47;
udv =3D uv – 
W= 47;
vdu
The integrals of some functions require using the integration by parts many times. 
This leads to special methods and formulas that summarize the results.
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Form= ulas From Integration by Parts
Integ= ration by parts:
Let u= =3D u(x), v =3D v(x), then
where= dv =3D v'dx and du =3D u'dx.
W= 47;
udv =3D uv – 
W= 47;
vdu
The integrals of some functions require using the integration by parts many times. 
This leads to special methods and formulas that summarize the results.
These formulas usually reduce the complexity of the integrands in stages until the answers are obtained.
------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0013_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1wAmAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQDX ACQAgAAAAAAAAAKujI+py+0Po5y02osV2Lz7D4biSJbmiabqyrac4cbyTNf2rcL4zvf+L9MBh8Si sSY8KpfMZrIJjUptz6n1ih1Vs9zudOsNi4vgsfl8K6PX7JW6DY+H3vK6nG7Pr/H6vpjvF5gFKFgo RWiYuISo2EjE6BjZAylZSRVgmbmIqdn5yOkZOgkqWnppiopEmsrqttoKa0IZizpLW2p7G5qr28nb m/kLXJlRbHyMnKy8jFEAADs= ------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0003.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Form= ulas From Integration by Parts
The T= ableau Method
------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0014.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Form= ulas From Integration by Parts
The T= ableau Method
Given=
+u
W= 47;
udv, set up the table as shown. 
dv
------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0014_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh1QF2AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAADU AXQAgAAAAAAAAAL/jI+py+0Po5y02ouz3rz7D4YiApTmiabqyrbuC8fyTNf2jef6zvf+DwwCDMKi 8YhMKpfMpvP5I0Kn1Kr1is1qn9Kt9wsOi8fkW7eMTqvX7Hbu7I7L5/Q6FW7P6/f8/grvFyg4SAgG WIiYqLgIdMj4CBkpaeI4aXmJqVeZydnpObb5KTpK6hRaipqqunO66voKq9IaS1uLOmubq4uJu+v7 q9gLPEzMJ1yMnOx2rNzsLMb8LD1tFU19ja1knc3d7bPtHS5OAz5ufi4bgL7O/qbeDh//Ui5fP01v n6+Mr98/zO8voC6AAgvGImgwoSqEChuOYugwIieIEitOomgxIyOM/xo7FuLoMaQfkCJL5iFpMqUc lCpbrmHpMiYZmDJrfqFpMycWnDp7TuHpMygToEKLHiFqNGmjd0qbrmTqNOpLqFKrzqRqNetNrFq7 7uTqNezPEWTLmj2LNq3atWzbuh0hNm41sHLrIkFqtyvevFn38q3q92/UwIKbEi6c9DDioooXB23s uCfkyDknU65p+XLMzJpbcu6c8jPokqJHhyxtuiPq1BlXs67o+nXE2LIb0q6d8Dbugrp3B+ztux/w 4PmGE69n/Hi85MrbMW++7jn0c9Knj6tuPRz27N22c8/m/fu18OLv0S3vlTx6Z+rX7zvv3mr7+MXm 0/8H/75T+/p98TPvP1B+ABr134C1FGjgQQIm6BOCDLri4IMLLSihTRFWWMqFGD70VocefghiiCKO SKKHBQAAOw== ------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0015.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Form= ulas From Integration by Parts
The T= ableau Method
Given=
+u
W= 47;
udv, set up the table as shown. 
differentiate
with alternate signs
̵= 1;u' 
+u''<= /span>
= 226;
̵= 1;u''' 
dv
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Form= ulas From Integration by Parts
The T= ableau Method
Given=
+u
W= 47;
udv, set up the table as shown. 
differentiate
with alternate signs
integrate
v
̵= 1;u' 
+u''<= /span>
∫ v dx
∫[∫ v dx]dx
= 226;
•
̵= 1;u''' 
dv
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Form= ulas From Integration by Parts
The T= ableau Method
Given=
+u
W= 47;
udv, set up the table as shown. 
differentiate
with alternate signs
integrate
v
̵= 1;u' 
+u''<= /span>
∫ v dx
∫[∫ v dx]dx
= 226;
•
̵= 1;u''' 
The integral  ∫ u dv is the sum = of the diagonal products. 
dv
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Form= ulas From Integration by Parts
The T= ableau Method
Given=
+u
W= 47;
udv, set up the table as shown. 
differentiate
with alternate signs
integrate
v
̵= 1;u' 
+u''<= /span>
∫ v dx
∫[∫ v dx]dx
= 226;
•
̵= 1;u''' 
The integral  ∫ u dv is the sum = of the diagonal products. 
dv
That = is, ∫ u dv =3D uv
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Form= ulas From Integration by Parts
The T= ableau Method
Given=
+u
W= 47;
udv, set up the table as shown. 
differentiate
with alternate signs
integrate
v
̵= 1;u' 
+u''<= /span>
∫ v dx
∫[∫ v dx]dx
= 226;
•
̵= 1;u''' 
The integral  ∫ u dv is the sum = of the diagonal products. 
dv
That = is, ∫ u dv =3D uv R= 11; u' ∫vdx
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Form= ulas From Integration by Parts
The T= ableau Method
Given=
+u
W= 47;
udv, set up the table as shown. 
differentiate
with alternate signs
integrate
v
̵= 1;u' 
+u''<= /span>
∫ v dx
∫[∫ v dx]dx
= 226;
•
̵= 1;u''' 
The integral  ∫ u dv is the sum = of the diagonal products. 
dv
That= is, ∫ u dv =3D uv R= 11; u' ∫vdx + u'' ∫[∫ v dx]dx ..
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Form= ulas From Integration by Parts
Example: Find ∫(x3 + 2x) ex/2 Slide 1
Form= ulas From Integration by Parts
Example: Find ∫(x3 + 2x) ex/2x3+2x
ex/2 
Set u =3D x3 + 2x, dv =3D ex/2dx Slide 1
Form= ulas From Integration by Parts
Example: Find ∫(x3 + 2x) ex/2x3+2x
differentiate
with alternate signs
ex/2 
Set u =3D x3 + 2x, dv =3D ex/2dx Slide 1
Form= ulas From Integration by Parts
Example: Find ∫(x3 + 2x) ex/2x3+2x
differentiate
with alternate signs
ex/2 
Set u =3D x3 + 2x, dv =3D ex/2dx–(3x2+2)
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Form= ulas From Integration by Parts
Example: Find ∫(x3 + 2x) ex/2x3+2x
differentiate
with alternate signs
6x
ex/2 
Set u =3D x3 + 2x, dv =3D ex/2dx–(3x2+2)
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Form= ulas From Integration by Parts
Example: Find ∫(x3 + 2x) ex/2x3+2x
differentiate
with alternate signs
6x
–6  
ex/2 
Set u =3D x3 + 2x, dv =3D ex/2dx–(3x2+2)
0 
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Form= ulas From Integration by Parts
Example: Find ∫(x3 + 2x) ex/2x3+2x
differentiate
with alternate signs
integr= ate
6x
–6  
ex/2 
Set u =3D x3 + 2x, dv =3D ex/2dx–(3x2+2)
0 
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Form= ulas From Integration by Parts
Example: Find ∫(x3 + 2x) ex/2x3+2x
differentiate
with alternate signs
integr= ate
6x
–6  
ex/2 
Set u =3D x3 + 2x, dv =3D ex/2dx–(3x2+2)
0 
2ex/2 
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Form= ulas From Integration by Parts
Example: Find ∫(x3 + 2x) ex/2x3+2x
differentiate
with alternate signs
integr= ate
6x
–6  
ex/2 
Set u =3D x3 + 2x, dv =3D ex/2dx–(3x2+2)
0 
2ex/2 
4ex/2 
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Form= ulas From Integration by Parts
Example: Find ∫(x3 + 2x) ex/2x3+2x
differentiate
with alternate signs
integr= ate
6x
–6  
ex/2 
Set u =3D x3 + 2x, dv =3D ex/2dx–(3x2+2)
0 
2ex/2 
4ex/2 
8ex/2 
16ex/2 
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Form= ulas From Integration by Parts
Example: Find ∫(x3 + 2x) ex/2x3+2x
differentiate
with alternate signs
integr= ate
6x
–6  
ex/2 
Set u =3D x3 + 2x, dv =3D ex/2dx–(3x2+2)
0 
2ex/2 
4ex/2 
8ex/2 
16ex/2 
Hence ∫(x3 + 2x) ex/2=3D
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Form= ulas From Integration by Parts
Example: Find ∫(x3 + 2x) ex/2x3+2x
differentiate
with alternate signs
integr= ate
6x
–6  
ex/2 
Set u =3D x3 + 2x, dv =3D ex/2dx–(3x2+2)
0 
2ex/2 
4ex/2 
8ex/2 
16ex/2 
Hence ∫(x3 + 2x) ex/2=3D
2ex/2(x3+2x)
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Form= ulas From Integration by Parts
Example: Find ∫(x3 + 2x) ex/2x3+2x
differentiate
with alternate signs
integr= ate
6x
–6  
ex/2 
Set u =3D x3 + 2x, dv =3D ex/2dx–(3x2+2)
0 
2ex/2 
4ex/2 
8ex/2 
16ex/2 
Hence ∫(x3 + 2x) ex/2=3D
2ex/2(x3+2x) – 4e<= /span>x/2(3x2+2)
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Form= ulas From Integration by Parts
Example: Find ∫(x3 + 2x) ex/2x3+2x
differentiate
with alternate signs
integr= ate
6x
–6  
ex/2 
Set u =3D x3 + 2x, dv =3D ex/2dx–(3x2+2)
0 
2ex/2 
4ex/2 
8ex/2 
16ex/2 
Hence ∫(x3 + 2x) ex/2=3D
2ex/2(x3+2x) – 4e<= /span>x/2(3x2+2) + 48xex/2 – 96ex/2 + c
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Form= ulas From Integration by Parts
Example: Find ∫(x3 + 2x) ex/2x3+2x
differentiate
with alternate signs
integr= ate
6x
–6  
ex/2 
Set u =3D x3 + 2x, dv =3D ex/2dx–(3x2+2)
0 
2ex/2 
4ex/2 
8ex/2 
16ex/2 
Hence ∫(x3 + 2x) ex/2=3D
2ex/2(x3+2x) – 4e<= /span>x/2(3x2+2) + 48xex/2 – 96ex/2 + c
=3D
ex/2 [2(x3+2x) – 4(3= x2+2) + 48x –= ; 96] + c
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Form= ulas From Integration by Parts
Example: Find ∫(x3 + 2x) ex/2x3+2x
differentiate
with alternate signs
integr= ate
6x
–6  
ex/2 
Set u =3D x3 + 2x, dv =3D ex/2dx–(3x2+2)
0 
2ex/2 
4ex/2 
8ex/2 
16ex/2 
Hence ∫(x3 + 2x) ex/2=3D
2ex/2(x3+2x) – 4e<= /span>x/2(3x2+2) + 48xex/2 – 96ex/2 + c
=3D
ex/2 [2(x3+2x) – 4(3= x2+2) + 48x –= ; 96] + c
=3D
ex/2 [2x3 – 12x2 + 52x – 1= 04] + c
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Form= ulas From Integration by Parts
The t= ableau method is used when the integrand is a product of polynomial with a functions that can be integrated repeatedly such as ex, sin(x), and cos(x).    <= /span>
------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0034.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Form= ulas From Integration by Parts
The tableau method is used when the integrand is a product of polynomial with a functions that can be integrated repeatedly such as ex, sin(x), and cos(x).    <= /span>
An= tiderivatives of trig-power
------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0035.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Form= ulas From Integration by Parts
The tableau method is used when the integrand is a product of polynomial with a functions that can be integrated repeatedly such as ex, sin(x), and cos(x).    <= /span>
An= tiderivatives of trig-power
From here on, we use s for sin(x), c for cos(x), and t for tan(x).
------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0058.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Form= ulas From Integration by Parts
The tableau method is used when the integrand is a product of polynomial with a functions that can be integrated repeatedly such as ex, sin(x), and cos(x).    <= /span>
An= tiderivatives of trig-power
From here on, we use s for sin(x), c for cos(x), and t for tan(x). Also we assume that n > 2.
------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0058_image093.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0036.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Form= ulas From Integration by Parts
The tableau method is used when the integrand is a product of polynomial with a functions that can be integrated repeatedly such as ex, sin(x), and cos(x).    <= /span>
An= tiderivatives of trig-power
&= #8747; cosn(x) dx
From here on, we use s for sin(x), c for cos(x), and t for tan(x). Also we assume that n > 2.
------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0036_image094.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0037.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Form= ulas From Integration by Parts
The tableau method is used when the integrand is a product of polynomial with a functions that can be integrated repeatedly such as ex, sin(x), and cos(x).    <= /span>
An= tiderivatives of trig-power
Writ= e ∫cndx =3D <= span style=3D'font-family:"";mso-ascii-font-family:"";mso-fareast-font-family:"= "; mso-bidi-font-family:"";font-size:178%'>∫cn–= ;1 cdx,&= #8747; cosn(x) dx
From here on, we use s for sin(x), c for cos(x), and t for tan(x). Also we assume that n > 2.
------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0037_image095.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0038.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Form= ulas From Integration by Parts
The tableau method is used when the integrand is a product of polynomial with a functions that can be integrated repeatedly such as ex, sin(x), and cos(x).    <= /span>
An= tiderivatives of trig-power
set = u =3D cn= –1        = ;            &n= bsp;  dv =3D cdx =
Writ= e ∫cndx =3D <= span style=3D'font-family:"";mso-ascii-font-family:"";mso-fareast-font-family:"= "; mso-bidi-font-family:"";font-size:178%'>∫cn–= ;1 cdx,&= #8747; cosn(x) dx
From here on, we use s for sin(x), c for cos(x), and t for tan(x). Also we assume that n > 2.
------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0038_image096.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C97CE5.7DBF3DD0 Content-Location: file:///C:/8F3256F3/17FormulasFromIntegrationbyParts_files/slide0039.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1