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In= tegrals of Log and Exponential Functions
------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0011.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
In= tegrals of Log and Exponential Functions
From the derivatives of log and exp functions,=   we obtain the following integrals:   <= /span>
------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0012.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
In= tegrals of Log and Exponential Functions
dx = =3D Ln(u) + c    
u=
u'
From the derivatives of log and exp functions,=   we obtain the following integrals:   <= /span>
W= 47;
W= 47;
u'eu=  dx = =3D eu=  + c=
------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0012_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0012_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhYgAgAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAACwBf ABQAgAAAAAAAAAKehI+pyxrR2otyUktzfNjA3QEfEoolyaHaaoanN6pHusTJ1L0tDZpybZuNdD7H J6bj8DK5ZFC4BNZgPxZz6AQRqQ5uy6rBKCva6wX7BG4tqbSot3Z7oFpl/Jske41MqnzMAsHzkuby FDXFJgcGY4NYdDN1s9UoxHj5prbI8Bi2iWnVWUFZB2pqSup5usra6voKGys7S1tre4url7sbWAAA Ow== ------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0013.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
In= tegrals of Log and Exponential Functions
dx = =3D Ln(u) + c    
u=
u'
From the derivatives of log and exp functions,=   we obtain the following integrals:   <= /span>
W= 47;
W= 47;
u'eu=  dx = =3D eu=  + c=
The integrand can be differed by a constant multiple and we use substitution method to carry out the integration.  
------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0013_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0014.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
In= tegrals of Log and Exponential Functions
dx = =3D Ln(u) + c    
u=
u'
From the derivatives of log and exp functions,=   we obtain the following integrals:   <= /span>
W= 47;
x2 + 1
x
W= 47;
u'eu=  dx = =3D eu=  + c=
The integrand can be differed by a constant multiple and we use substitution method to carry out the integration.  
Example: a. Integrate
∫
dx
------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0014_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhSQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQBI AAEAgAAAAAAAAAIHjI+py+2vCgA7 ------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0014_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0015.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
In= tegrals of Log and Exponential Functions
dx = =3D Ln(u) + c    
u=
u'
From the derivatives of log and exp functions,=   we obtain the following integrals:   <= /span>
W= 47;
x2 + 1
x
W= 47;
u'eu=  dx = =3D eu=  + c=
The integrand can be differed by a constant multiple and we use substitution method to carry out the integration.  
Example: a. Integrate
∫
dx
Set u =3D x2 + 1
sub-method
------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0015_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhSQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQBI AAEAgAAAAAAAAAIHjI+py+2vCgA7 ------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0015_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0015_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhygBOAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQDJ AEwAgAAAAAAAAAL/jI+py+0Po5y02guB3rz7D4biSJbmiabqGhrsC8fyTNeja+f6zvcn7gsKh8QU sIhMKn3HpfMJNQai1Kq107xqt8Qs9wu2ecPksmpsTqtB6LV73X7LyfG5fVu/66n5vX/Z9yfYNTVo GBV4qJiTuOgo0/gouRI5aWlSeanZUrjpCZP5KRoq6klaqnmKaqm6Ktnq6ggbqzhLa2h7K5ir68fb q/cLbCc8LFds7IacrLbMbOb8TNcpPUpdbXqNnaq9zdrt/QoeLjtOXmt+jpuuvsve7vsOHyw/T1xv f4yfr7zP3+zvH7SAAqcVXHdwULSEQxYyDOLwYY+IEndQrMiIIMYqJxc30ujoEZLGkE9Aknxh8iSl kSqTpGyJ4iVMTBhq2ryJM6fOnREKAAA7 ------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0016.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
In= tegrals of Log and Exponential Functions
dx = =3D Ln(u) + c    
u=
u'
From the derivatives of log and exp functions,=   we obtain the following integrals:   <= /span>
W= 47;
x2 + 1
x
W= 47;
u'eu=  dx = =3D eu=  + c=
The integrand can be differed by a constant multiple and we use substitution method to carry out the integration.  
Example: a. Integrate
∫
dx
Set u =3D x2 + 1
dx
du
=3D 2x 
sub-method
------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0016_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhSQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQBI AAEAgAAAAAAAAAIHjI+py+2vCgA7 ------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0016_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0016_image011.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0016_image012.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhygBOAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQDJ AEwAgAAAAAAAAAL/jI+py+0Po5y02guB3rz7D4biSJbmiabqGhrsC8fyTNeja+f6zvcn7gsKh8QU sIhMKn3HpfMJNQai1Kq107xqt8Qs9wu2ecPksmpsTqtB6LV73X7LyfG5fVu/66n5vX/Z9yfYNTVo GBV4qJiTuOgo0/gouRI5aWlSeanZUrjpCZP5KRoq6klaqnmKaqm6Ktnq6ggbqzhLa2h7K5ir68fb q/cLbCc8LFds7IacrLbMbOb8TNcpPUpdbXqNnaq9zdrt/QoeLjtOXmt+jpuuvsve7vsOHyw/T1xv f4yfr7zP3+zvH7SAAqcVXHdwULSEQxYyDOLwYY+IEndQrMiIIMYqJxc30ujoEZLGkE9Aknxh8iSl kSqTpGyJ4iVMTBhq2ryJM6fOnREKAAA7 ------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0017.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
In= tegrals of Log and Exponential Functions
dx = =3D Ln(u) + c    
u=
u'
From the derivatives of log and exp functions,=   we obtain the following integrals:   <= /span>
W= 47;
x2 + 1
x
W= 47;
u'eu=  dx = =3D eu=  + c=
The integrand can be differed by a constant multiple and we use substitution method to carry out the integration.  
Example: a. Integrate
∫
dx
Set u =3D x2 + 1
dx
du
=3D 2x  or
dx
=3D
2x
du
sub-method
------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0017_image013.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhSQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQBI AAEAgAAAAAAAAAIHjI+py+2vCgA7 ------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0017_image014.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0017_image015.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0017_image016.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhygBOAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAADJ AEwAgAAAAAAAAAL/jI+py+0Po5y02guB3rz7D4biSJbmiabqGhrsC8fyTNeja+f6zvcn7gsKh8QU sIhMKn3HpfMJNQai1Kq107xqt8Qs9wu2ecPksmpsTqtB6LV73X7LyfG5fVu/66n5vX/Z9yfYNTVo GBV4qJiTuOgo0/gouRI5aWlSeanZUrjpCZP5KRoq6klaqnmKaqm6Ktnq6ggbqzhLa2h7K5ir68fb q/cLbCc8LFds7IacrLbMbOb8TNcpPUpdbXqNnaq9zdrt/QoeLjtODohRF31etc6OaP6+Gy/vS18f fI9PrL9/3O9PGcCAzQYShGbw4DSFg9wxDOLwYY+IEndQrMgoIcYrKBc30ujoEZLGkPBI/jP5BiRK SiNXKlHpEgXMmJjS2byJM6fOnTwTFAAAOw== ------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0018.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
In= tegrals of Log and Exponential Functions
dx = =3D Ln(u) + c    
u=
u'
From the derivatives of log and exp functions,=   we obtain the following integrals:   <= /span>
W= 47;
x2 + 1
x
W= 47;
u'eu=  dx = =3D eu=  + c=
The integrand can be differed by a constant multiple and we use substitution method to carry out the integration.  
Example: a. Integrate
∫
dx
Set u =3D x2 + 1
dx
du
=3D 2x  or
dx
=3D
2x
du
sub-method
u
x
∫
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In= tegrals of Log and Exponential Functions
dx = =3D Ln(u) + c    
u=
u'
From the derivatives of log and exp functions,=   we obtain the following integrals:   <= /span>
W= 47;
x2 + 1
x
W= 47;
u'eu=  dx = =3D eu=  + c=
The integrand can be differed by a constant multiple and we use substitution method to carry out the integration.  
Example: a. Integrate
∫
dx
Set u =3D x2 + 1
dx
du
=3D 2x  or
dx
=3D
2x
du
sub-method
u
x
∫
 =3D
2x
du
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In= tegrals of Log and Exponential Functions
dx = =3D Ln(u) + c    
u=
u'
From the derivatives of log and exp functions,=   we obtain the following integrals:   <= /span>
W= 47;
x2 + 1
x
W= 47;
u'eu=  dx = =3D eu=  + c=
The integrand can be differed by a constant multiple and we use substitution method to carry out the integration.  
Example: a. Integrate
∫
dx
Set u =3D x2 + 1
dx
du
=3D 2x  or
dx
=3D
2x
du
sub-method
u
x
∫
 =3D
2x
du
2
1
u
1
∫
du
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In= tegrals of Log and Exponential Functions
dx = =3D Ln(u) + c    
u=
u'
From the derivatives of log and exp functions,=   we obtain the following integrals:   <= /span>
W= 47;
x2 + 1
x
W= 47;
u'eu=  dx = =3D eu=  + c=
The integrand can be differed by a constant multiple and we use substitution method to carry out the integration.  
Example: a. Integrate
∫
dx
Set u =3D x2 + 1
dx
du
=3D 2x  or
dx
=3D
2x
du
sub-method
u
x
∫
 =3D
2x
du
2
1
u
1
∫
du
 =3D Ln(u) + c
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In= tegrals of Log and Exponential Functions
dx = =3D Ln(u) + c    
u=
u'
From the derivatives of log and exp functions,=   we obtain the following integrals:   <= /span>
W= 47;
x2 + 1
x
W= 47;
u'eu=  dx = =3D eu=  + c=
The integrand can be differed by a constant multiple and we use substitution method to carry out the integration.  
Example: a. Integrate
∫
dx
Set u =3D x2 + 1
dx
du
=3D 2x  or
dx
=3D
2x
du
sub-method
u
x
∫
 =3D
2x
du
2
1
u
1
∫
du
 =3D Ln(u) + c =3D Ln(x2 + 1) + c
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In= tegrals of Log and Exponential Functions
Example: b. Integrate
∫
sec2(x)etan(x) dx
------=_NextPart_01C97CE4.F8D3FD40 Content-Location: file:///C:/4D764A33/11IntegralsofLogandExpfunctions_files/slide0021.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
In= tegrals of Log and Exponential Functions
Example: b. Integrate
∫
Set u =3D tan(x)
sec2(x)etan(x) dx
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In= tegrals of Log and Exponential Functions
Example: b. Integrate
∫
Set u =3D tan(x)
dx
du
=3D sec2(x)
sec2(x)etan(x) dx
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In= tegrals of Log and Exponential Functions
Example: b. Integrate
∫
Set u =3D tan(x)
dx
du
=3D sec2(x)
dx
=3D
sec2(x)
du
sec2(x)etan(x) dx
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In= tegrals of Log and Exponential Functions
Example: b. Integrate
∫
Set u =3D tan(x)
dx
du
=3D sec2(x)
dx
=3D
sec2(x)
du
sec2(x)etan(x) dx
∫
sec2(x)eu
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In= tegrals of Log and Exponential Functions
Example: b. Integrate
∫
Set u =3D tan(x)
dx
du
=3D sec2(x)
dx
=3D
sec2(x)
du
sec2(x)etan(x) dx
∫
sec2(x)eu
sec2(x)
du
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In= tegrals of Log and Exponential Functions
Example: b. Integrate
∫
Set u =3D tan(x)
dx
du
=3D sec2(x)
dx
=3D
sec2(x)
du
sec2(x)etan(x) dx
∫
sec2(x)eu
sec2(x)
du
=3D
∫
 eu
du
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In= tegrals of Log and Exponential Functions
Example: b. Integrate
∫
Set u =3D tan(x)
dx
du
=3D sec2(x)
dx
=3D
sec2(x)
du
sec2(x)etan(x) dx
∫
sec2(x)eu
sec2(x)
du
=3D
∫
 eu
du
=3D eu + c
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In= tegrals of Log and Exponential Functions
Example: b. Integrate
∫
Set u =3D tan(x)
dx
du
=3D sec2(x)
dx
=3D
sec2(x)
du
sec2(x)etan(x) dx
∫
sec2(x)eu
sec2(x)
du
=3D
∫
 eu
du
=3D eu + c =3D etan(x)=  + c
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In= tegrals of Log and Exponential Functions
Example: b. Integrate
∫
Set u =3D tan(x)
dx
du
=3D sec2(x)
dx
=3D
sec2(x)
du
sec2(x)etan(x) dx
∫
sec2(x)eu
sec2(x)
du
=3D
∫
 eu
du
=3D eu + c =3D etan(x)=  + c
c. Integrate
∫
x
Ln(x)
dx
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In= tegrals of Log and Exponential Functions
Example: b. Integrate
∫
Set u =3D tan(x)
dx
du
=3D sec2(x)
dx
=3D
sec2(x)
du
sec2(x)etan(x) dx
∫
sec2(x)eu
sec2(x)
du
=3D
∫
 eu
du
=3D eu + c =3D etan(x)=  + c
c. Integrate
∫
Set u =3D Ln(x)
x
Ln(x)
dx
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In= tegrals of Log and Exponential Functions
Example: b. Integrate
∫
Set u =3D tan(x)
dx
du
=3D sec2(x)
dx
=3D
sec2(x)
du
sec2(x)etan(x) dx
∫
sec2(x)eu
sec2(x)
du
=3D
∫
 eu
du
=3D eu + c =3D etan(x)=  + c
c. Integrate
∫
Set u =3D Ln(x)
dx
du
=3D
x
Ln(x)
dx
x
1
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In= tegrals of Log and Exponential Functions
Example: b. Integrate
∫
Set u =3D tan(x)
dx
du
=3D sec2(x)
dx
=3D
sec2(x)
du
sec2(x)etan(x) dx
∫
sec2(x)eu
sec2(x)
du
=3D
∫
 eu
du
=3D eu + c =3D etan(x)=  + c
c. Integrate
∫
Set u =3D Ln(x)
dx
du
=3D
dx
=3D
x du
x
Ln(x)
dx
x
1
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In= tegrals of Log and Exponential Functions
Example: b. Integrate
∫
Set u =3D tan(x)
dx
du
=3D sec2(x)
dx
=3D
sec2(x)
du
sec2(x)etan(x) dx
∫
sec2(x)eu
sec2(x)
du
=3D
∫
 eu
du