MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C97CE7.B7A456A0" This document is a Web archive file. If you are seeing this message, this means your browser or editor doesn't support Web archive files. For more information on the Web archive format, go to http://officeupdate.microsoft.com/office/webarchive.htm ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1

This presentation contains content that your browser may not be able to = show properly. This presentation was optimized for more recent versions of Micro= soft Internet Explorer.

If you would like to proceed anyway, click here.<= /p> ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/master03.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii"

Click to edit Master title style
Click to edit Master text styles
Second level
Third level
Fourth level
Fifth level
‹date/time›
‹footer›
‹#›
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/master03.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/preview.wmf Content-Transfer-Encoding: base64 Content-Type: image/x-wmf AQAJAAAD+AAAAAYASAAAAAAABAAAAAMBCAAFAAAACwIAAAAABQAAAAwC7wLpAwMAAAAeAAcAAAD8 AgAA////AAAABAAAAC0BAAAIAAAA+gIFAAAAAAD///8ABAAAAC0BAQAMAAAAQAkhAPAAAAAAAAAA 7wLpA/////8IAAAA+gIAAAAAAAAAAAAABAAAAC0BAgAHAAAA/AIAAP///wAAAAQAAAAtAQMABAAA ACcB//8cAAAA+wLU/wAAAAAAAJABAAAAAABAACJBcmlhbAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAQAAAAtAQQABAAAAC4BGAAEAAAAAgEBAAUAAAAJAgAAAAJIAAAAMgo5ADwAKwAAAEludGVn cmFscyBvZiBQcm9kdWN0cyBvZiBUYW5nZW50IGFuZCBTZWNhbnQADAAYAA0AGAAZAA8AGAAKABYA DQAYAA0ADAAdABAAGQAYABgAFwAMABYADQAYAA0ADQAaABkAGAAZABkAGAANAAwAGQAZABgADQAd ABgAFwAZABgADAAEAAAALgEAABwAAAD7AhAABwAAAAAAvAIAAAAAAQICIlN5c3RlbQAAAAAAABgA AABoaRMAAgAAAOQEAAAAAAAABAAAAC0BBQAEAAAA8AEEAAMAAAAAAA== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0002.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Integ= rals of Products of Tangent and Secant
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0019.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
In this section we write t =3D t(x) =3Dtan(x) and
e =3D e(x) =3D sec(x). 
Integ= rals of Products of Tangent and Secant
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0066.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
In this section we write t =3D t(x) =3Dtan(x) and
e =3D e(x) =3D sec(x).  They satisfy
        = ;      e22 + 1 or t2 =3D e – 1
Integ= rals of Products of Tangent and Secant
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0067.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
In this section we write t =3D t(x) =3Dtan(x) and
e =3D e(x) =3D sec(x).  They satisfy
        = ;      e22 + 1 or t2 =3D e – 1
Integ= rals of Products of Tangent and Secant
Their derivatives are =
        t'(x) =3D e2 Slide 1
In this section we write t =3D t(x) =3Dtan(x) and
e =3D e(x) =3D sec(x).  They satisfy
        = ;      e22 + 1 or t2 =3D e – 1
Integ= rals of Products of Tangent and Secant
Their derivatives are =
        t'(x) =3D e2 Slide 1
integrals of tn  and en for n > 2.
In this section we write t =3D t(x) =3Dtan(x) and
e =3D e(x) =3D sec(x).  They satisfy
        = ;      e22 + 1 or t2 =3D e – 1
Integ= rals of Products of Tangent and Secant
Their derivatives are =
        t'(x) =3D e2 Slide 1
integrals of tn  and en for n > 2.
In this section we write t =3D t(x) =3Dtan(x) and
e =3D e(x) =3D sec(x).  They satisfy
        = ;      e22 + 1 or t2 =3D e – 1
Integ= rals of Products of Tangent and Secant
Their derivatives are =
        t'(x) =3D e2 ∫tndx =3D        = ;   –  ∫tn–2dx
n = 211; 1
tn= –1
 ∫endx =3D        = ;   +        = ;  ∫edx
n = 211; 1
en= –2t
n–= ;1
n̵= 1;2
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0070_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEAADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0070_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhXQE4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABc ATYAgAAAAAAAAAL/jI+py+0Po5y02ouz3rxzAIbiSJbmiabqyrbuC8fyTNf2PRv4zvf+DwwKh8SR rohMKpfMpjMXeEqn1Kr1ijpit9yu9/vSgsfksrkpPqvX7HY46o7L5+00/Y7PT+36vv/vwwc4SFi4 ImiYqFiIuOj4eNcIOUl5JlmJmbl1qdnpiQb3KTrqxEl6impjmsrayrLqGisrAjtr21p7q0uau+vb 2fsrTBk8bLxYfKxMmLzs3Ncc5zEd+kwXbZ0tXa3dnYjtHU4GLl7eRW6eboWu3v7E7h6vBC9fP0Rv nx/Ird+Phe8vIBSBBKsALIhQxcGEDEssbAgRxMOIDSdSTGjxYsGMIRoFcuzo7yNIfSJH2itpUh7K lO6ouXwJM6bMmTRr2uRQAAA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0070_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhXQE4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABc ATYAgAAAAAAAAAL/jI+py+0Po5y02ouz3rxzAIbiSJbmiabqyrbuC8fyTNf2PRv4zvf+DwwKh8SR rohMKpfMpjMXeEqn1Kr1ijpit9yu9/vSgsfksrkpPqvX7HY46o7L5+00/Y7PT+36vv/vwwc4SFi4 ImiYqFiIuOj4eNcIOUl5JlmJmbl1qdnpiQb3KTrqxEl6impjmsrayrLqGisrAjtr21p7q0uau+vb 2fsrTBk8bLxY/OeRceyX3AytHBpNjflcjS13nc29tt0NTvYdTt41Xo5udZ7O/rTeDq/0Hk8/NF+P Hzidz7+53w+Qyr2ABN8UPOjuH8KF9hQyfKgPosQgAyc+rGhxIcaMIAc3ciTo8SPAkCL5kSyJ7yRK estaunwJM6bMmTRrziwAADs= ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0070_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhNwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA2 AAEAgAAAAAAAAAIGjI+py90FADs= ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0062.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
and secant functions.           = ;  
Integ= rals of Products of Tangent and Secant
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0071.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
and secant functions.           = ;  
W= 47;t(x) dx =3D ∫
Integ= rals of Products of Tangent and Secant
s(x)=
c(x)=
dx <= /span>
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0071_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0072.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
and secant functions.           = ;  
W= 47;t(x) dx =3D ∫
Integ= rals of Products of Tangent and Secant
s(x)=
c(x)=
dx <= /span>
substitution
u =3D c(x) 
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0072_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegCQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQB5 AI8AgAAAAAAAAAL/jI+py+0Po5wG2Iuz3rz7D4aiWI3miaZqWK7uC8dWK9f2rdH4zrt6DwyCfsKi cRY4KpXEpfPWfEph0ak1Vb1qScmtF9v9irnj8jBsTmey6i+7vX3Dr/L5tG5/4vPLPf/o91cUKBhE WNhziLijuAiF5mjVGCkzSUkFeelkqbnC2QkGKvUpakJaSobKlKkKdNrq8QrLITubw2r7mGuIu1vZ 64sZzFMbXOx7vJucu2zbPPsMG906rVqNel2aLboN2t35rRl+OU5ZHnnumL64jthe+C4Y/zfPV593 b5c/tw/X3/ZPTcA0A80ULHNwTEIxC90AG+bpIcRQE2009HIxjsSKhKY2ckz1UVhIkSN9eCzZIaMW lXROotzAUpLLl2tm0rwQ847NmwByjtp506ceoDSFbiL60mgfpCiVruKJwikgpiWlGrE6iOpIrEK4 8oJ6wqsrrSHFJiL70SwxtBzVMmJb0S0OubrAjqBrEe5EvDX4/rJ7Vy9EvzEIkwT8gYLixYwbO35Q AAA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0072_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0080.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
and secant functions.           = ;  
W= 47;t(x) dx =3D ∫
Integ= rals of Products of Tangent and Secant
s(x)=
c(x)=
dx <= /span>
substitution
u =3D c(x) 
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0080_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegCQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQB5 AI8AgAAAAAAAAAL/jI+py+0Po5wG2Iuz3rz7D4aiWI3miaZqWK7uC8dWK9f2rdH4zrt6DwyCfsKi cRY4KpXEpfPWfEph0ak1Vb1qScmtF9v9irnj8jBsTmey6i+7vX3Dr/L5tG5/4vPLPf/o91cUKBhE WNhziLijuAiF5mjVGCkzSUkFeelkqbnC2QkGKvUpakJaSobKlKkKdNrq8QrLITubw2r7mGuIu1vZ 64sZzFMbXOx7vJucu2zbPPsMG906rVqNel2aLboN2t35rRl+OU5ZHnnumL64jthe+C4Y/zfPV593 b5c/tw/X3/ZPTcA0A80ULHNwTEIxC90AG+bpIcRQE2009HIxjsSKhKY2ckz1UVhIkSN9eCzZIaMW lXROotzAUpLLl2tm0rwQ847NmwByjtp506ceoDSFbiL60mgfpCiVruKJwikgpiWlGrE6iOpIrEK4 8oJ6wqsrrSHFJiL70SwxtBzVMmJb0S0OubrAjqBrEe5EvDX4/rJ7Vy9EvzEIkwT8gYLixYwbO35Q AAA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0080_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0128.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
and secant functions.           = ;  
W= 47;t(x) dx =3D ∫
Integ= rals of Products of Tangent and Secant
s(x)=
c(x)=
dx <= /span>
substitution
u =3D c(x)  du
 dx=3D –s(x)
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0128_image011.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegCQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQB5 AI8AgAAAAAAAAAL/jI+py+0Po5wG2Iuz3rz7D4aiWI3miaZqWK7uC8dWK9f2rdH4zrt6DwyCfsKi cRY4KpXEpfPWfEph0ak1Vb1qScmtF9v9irnj8jBsTmey6i+7vX3Dr/L5tG5/4vPLPf/o91cUKBhE WNhziLijuAiF5mjVGCkzSUkFeelkqbnC2QkGKvUpakJaSobKlKkKdNrq8QrLITubw2r7mGuIu1vZ 64sZzFMbXOx7vJucu2zbPPsMG906rVqNel2aLboN2t35rRl+OU5ZHnnumL64jthe+C4Y/zfPV593 b5c/tw/X3/ZPTcA0A80ULHNwTEIxC90AG+bpIcRQE2009HIxjsSKhKY2ckz1UVhIkSN9eCzZIaMW lXROotzAUpLLl2tm0rwQ847NmwByjtp506ceoDSFbiL60mgfpCiVruKJwikgpiWlGrE6iOpIrEK4 8oJ6wqsrrSHFJiL70SwxtBzVMmJb0S0OubrAjqBrEe5EvDX4/rJ7Vy9EvzEIkwT8gYLixYwbO35Q AAA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0128_image012.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0128_image013.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAb AAEAgAAAAAAAAAIEjI+pWgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0129.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
and secant functions.           = ;  
W= 47;t(x) dx =3D ∫
Integ= rals of Products of Tangent and Secant
s(x)=
c(x)=
dx <= /span>
substitution
u =3D c(x)  du
 dx=3D –s(x)
R= 11;s(x)
dx = =3D
du
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0129_image014.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegCQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQB5 AI8AgAAAAAAAAAL/jI+py+0Po5wG2Iuz3rz7D4aiWI3miaZqWK7uC8dWK9f2rdH4zrt6DwyCfsKi cRY4KpXEpfPWfEph0ak1Vb1qScmtF9v9irnj8jBsTmey6i+7vX3Dr/L5tG5/4vPLPf/o91cUKBhE WNhziLijuAiF5mjVGCkzSUkFeelkqbnC2QkGKvUpakJaSobKlKkKdNrq8QrLITubw2r7mGuIu1vZ 64sZzFMbXOx7vJucu2zbPPsMG906rVqNel2aLboN2t35rRl+OU5ZHnnumL64jthe+C4Y/zfPV593 b5c/tw/X3/ZPTcA0A80ULHNwTEIxC90AG+bpIcRQE2009HIxjsSKhKY2ckz1UVhIkSN9eCzZIaMW lXROotzAUpLLl2tm0rwQ847NmwByjtp506ceoDSFbiL60mgfpCiVruKJwikgpiWlGrE6iOpIrEK4 8oJ6wqsrrSHFJiL70SwxtBzVMmJb0S0OubrAjqBrEe5EvDX4/rJ7Vy9EvzEIkwT8gYLixYwbO35Q AAA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0129_image015.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0129_image016.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0129_image017.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAb AAEAgAAAAAAAAAIEjI+pWgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0130.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
and secant functions.           = ;  
W= 47;t(x) dx =3D ∫
Integ= rals of Products of Tangent and Secant
s(x)=
c(x)=
dx
substitution
u = =3D c(x) 
du
 dx
=3D –s(x)
=3D = ∫
s(x)=
udu
–s(x)
R= 11;s(x)
dx = =3D
du
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0130_image018.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegCQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQB5 AI8AgAAAAAAAAAL/jI+py+0Po5wG2Iuz3rz7D4aiWI3miaZqWK7uC8dWK9f2rdH4zrt6DwyCfsKi cRY4KpXEpfPWfEph0ak1Vb1qScmtF9v9irnj8jBsTmey6i+7vX3Dr/L5tG5/4vPLPf/o91cUKBhE WNhziLijuAiF5mjVGCkzSUkFeelkqbnC2QkGKvUpakJaSobKlKkKdNrq8QrLITubw2r7mGuIu1vZ 64sZzFMbXOx7vJucu2zbPPsMG906rVqNel2aLboN2t35rRl+OU5ZHnnumL64jthe+C4Y/zfPV593 b5c/tw/X3/ZPTcA0A80ULHNwTEIxC90AG+bpIcRQE2009HIxjsSKhKY2ckz1UVhIkSN9eCzZIaMW lXROotzAUpLLl2tm0rwQ847NmwByjtp506ceoDSFbiL60mgfpCiVruKJwikgpiWlGrE6iOpIrEK4 8oJ6wqsrrSHFJiL70SwxtBzVMmJb0S0OubrAjqBrEe5EvDX4/rJ7Vy9EvzEIkwT8gYLixYwbO35Q AAA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0130_image019.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0130_image020.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhWwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABa AAEAgAAAAAAAAAIKjI+pywkPX5uUFgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0130_image021.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0130_image022.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAb AAEAgAAAAAAAAAIEjI+pWgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0131.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
and secant functions.           = ;  
W= 47;t(x) dx =3D ∫
Integ= rals of Products of Tangent and Secant
s(x)=
c(x)=
dx <= /span>
substitution
u =3D c(x)  du
 dx=3D –s(x)
=3D = ∫
s(x)=
u
du
–= s(x)
=3D = ∫
–= 1
u
du
R= 11;s(x)
dx = =3D
du
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0131_image023.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegCQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQB5 AI8AgAAAAAAAAAL/jI+py+0Po5wG2Iuz3rz7D4aiWI3miaZqWK7uC8dWK9f2rdH4zrt6DwyCfsKi cRY4KpXEpfPWfEph0ak1Vb1qScmtF9v9irnj8jBsTmey6i+7vX3Dr/L5tG5/4vPLPf/o91cUKBhE WNhziLijuAiF5mjVGCkzSUkFeelkqbnC2QkGKvUpakJaSobKlKkKdNrq8QrLITubw2r7mGuIu1vZ 64sZzFMbXOx7vJucu2zbPPsMG906rVqNel2aLboN2t35rRl+OU5ZHnnumL64jthe+C4Y/zfPV593 b5c/tw/X3/ZPTcA0A80ULHNwTEIxC90AG+bpIcRQE2009HIxjsSKhKY2ckz1UVhIkSN9eCzZIaMW lXROotzAUpLLl2tm0rwQ847NmwByjtp506ceoDSFbiL60mgfpCiVruKJwikgpiWlGrE6iOpIrEK4 8oJ6wqsrrSHFJiL70SwxtBzVMmJb0S0OubrAjqBrEe5EvDX4/rJ7Vy9EvzEIkwT8gYLixYwbO35Q AAA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0131_image024.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0131_image025.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0131_image026.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhWwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABa AAEAgAAAAAAAAAIKjI+pywkPX5uUFgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0131_image027.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0131_image028.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAb AAEAgAAAAAAAAAIEjI+pWgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0132.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
and secant functions.           = ;  
W= 47;t(x) dx =3D ∫
Integ= rals of Products of Tangent and Secant
s(x)=
c(x)=
dx <= /span>
substitution
u =3D c(x)  du
 dx=3D –s(x)
=3D = ∫
s(x)=
u
du
–= s(x)
=3D = ∫
–= 1
u
du
=3D<= /span> R= 11;Ln(|u|) + k
R= 11;s(x)
dx = =3D
du
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0132_image029.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegCQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQB5 AI8AgAAAAAAAAAL/jI+py+0Po5wG2Iuz3rz7D4aiWI3miaZqWK7uC8dWK9f2rdH4zrt6DwyCfsKi cRY4KpXEpfPWfEph0ak1Vb1qScmtF9v9irnj8jBsTmey6i+7vX3Dr/L5tG5/4vPLPf/o91cUKBhE WNhziLijuAiF5mjVGCkzSUkFeelkqbnC2QkGKvUpakJaSobKlKkKdNrq8QrLITubw2r7mGuIu1vZ 64sZzFMbXOx7vJucu2zbPPsMG906rVqNel2aLboN2t35rRl+OU5ZHnnumL64jthe+C4Y/zfPV593 b5c/tw/X3/ZPTcA0A80ULHNwTEIxC90AG+bpIcRQE2009HIxjsSKhKY2ckz1UVhIkSN9eCzZIaMW lXROotzAUpLLl2tm0rwQ847NmwByjtp506ceoDSFbiL60mgfpCiVruKJwikgpiWlGrE6iOpIrEK4 8oJ6wqsrrSHFJiL70SwxtBzVMmJb0S0OubrAjqBrEe5EvDX4/rJ7Vy9EvzEIkwT8gYLixYwbO35Q AAA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0132_image030.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0132_image031.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0132_image032.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhWwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABa AAEAgAAAAAAAAAIKjI+pywkPX5uUFgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0132_image033.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0132_image034.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAb AAEAgAAAAAAAAAIEjI+pWgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0133.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
and secant functions.           = ;  
W= 47;t(x) dx =3D ∫
Integ= rals of Products of Tangent and Secant
s(x)=
c(x)=
dx <= /span>
substitution
u =3D c(x)  du
 dx=3D –s(x)
=3D = ∫
s(x)=
u
du
–= s(x)
=3D = ∫
–= 1
u
du
=3D<= /span> R= 11;Ln(|u|) + k
=3D<= /span> R= 11;Ln(|c(x)|) + k
R= 11;s(x)
dx = =3D
du
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0133_image035.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegCQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQB5 AI8AgAAAAAAAAAL/jI+py+0Po5wG2Iuz3rz7D4aiWI3miaZqWK7uC8dWK9f2rdH4zrt6DwyCfsKi cRY4KpXEpfPWfEph0ak1Vb1qScmtF9v9irnj8jBsTmey6i+7vX3Dr/L5tG5/4vPLPf/o91cUKBhE WNhziLijuAiF5mjVGCkzSUkFeelkqbnC2QkGKvUpakJaSobKlKkKdNrq8QrLITubw2r7mGuIu1vZ 64sZzFMbXOx7vJucu2zbPPsMG906rVqNel2aLboN2t35rRl+OU5ZHnnumL64jthe+C4Y/zfPV593 b5c/tw/X3/ZPTcA0A80ULHNwTEIxC90AG+bpIcRQE2009HIxjsSKhKY2ckz1UVhIkSN9eCzZIaMW lXROotzAUpLLl2tm0rwQ847NmwByjtp506ceoDSFbiL60mgfpCiVruKJwikgpiWlGrE6iOpIrEK4 8oJ6wqsrrSHFJiL70SwxtBzVMmJb0S0OubrAjqBrEe5EvDX4/rJ7Vy9EvzEIkwT8gYLixYwbO35Q AAA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0133_image036.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0133_image037.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0133_image038.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhWwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABa AAEAgAAAAAAAAAIKjI+pywkPX5uUFgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0133_image039.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0133_image040.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAb AAEAgAAAAAAAAAIEjI+pWgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0136.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
and secant functions.           = ;  
W= 47;t(x) dx =3D ∫
Integ= rals of Products of Tangent and Secant
s(x)=
c(x)=
dx <= /span>
substitution
u =3D c(x)  du
 dx=3D –s(x)
=3D = ∫
s(x)=
u
du
–= s(x)
=3D = ∫
–= 1
u
du
=3D<= /span> R= 11;Ln(|u|) + k
=3D<= /span> R= 11;Ln(|c(x)|) + k
=3D<= /span> 1
|c(x= )|
R= 11;s(x)
dx = =3D
du
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0136_image041.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegCQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQB5 AI8AgAAAAAAAAAL/jI+py+0Po5wG2Iuz3rz7D4aiWI3miaZqWK7uC8dWK9f2rdH4zrt6DwyCfsKi cRY4KpXEpfPWfEph0ak1Vb1qScmtF9v9irnj8jBsTmey6i+7vX3Dr/L5tG5/4vPLPf/o91cUKBhE WNhziLijuAiF5mjVGCkzSUkFeelkqbnC2QkGKvUpakJaSobKlKkKdNrq8QrLITubw2r7mGuIu1vZ 64sZzFMbXOx7vJucu2zbPPsMG906rVqNel2aLboN2t35rRl+OU5ZHnnumL64jthe+C4Y/zfPV593 b5c/tw/X3/ZPTcA0A80ULHNwTEIxC90AG+bpIcRQE2009HIxjsSKhKY2ckz1UVhIkSN9eCzZIaMW lXROotzAUpLLl2tm0rwQ847NmwByjtp506ceoDSFbiL60mgfpCiVruKJwikgpiWlGrE6iOpIrEK4 8oJ6wqsrrSHFJiL70SwxtBzVMmJb0S0OubrAjqBrEe5EvDX4/rJ7Vy9EvzEIkwT8gYLixYwbO35Q AAA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0136_image042.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0136_image043.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0136_image044.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhWwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABa AAEAgAAAAAAAAAIKjI+pywkPX5uUFgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0136_image045.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0136_image046.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0136_image047.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAb AAEAgAAAAAAAAAIEjI+pWgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0134.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
and secant functions.           = ;  
W= 47;t(x) dx =3D ∫
Integ= rals of Products of Tangent and Secant
s(x)=
c(x)=
dx <= /span>
substitution
u =3D c(x)  du
 dx=3D –s(x)
=3D = ∫
s(x)=
u
du
–= s(x)
=3D = ∫
–= 1
u
du
=3D<= /span> R= 11;Ln(|u|) + k
=3D<= /span> R= 11;Ln(|c(x)|) + k
=3D<= /span> 1
|c(x= )|
R= 11;s(x)
dx = =3D
du
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0134_image048.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegCQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQB5 AI8AgAAAAAAAAAL/jI+py+0Po5wG2Iuz3rz7D4aiWI3miaZqWK7uC8dWK9f2rdH4zrt6DwyCfsKi cRY4KpXEpfPWfEph0ak1Vb1qScmtF9v9irnj8jBsTmey6i+7vX3Dr/L5tG5/4vPLPf/o91cUKBhE WNhziLijuAiF5mjVGCkzSUkFeelkqbnC2QkGKvUpakJaSobKlKkKdNrq8QrLITubw2r7mGuIu1vZ 64sZzFMbXOx7vJucu2zbPPsMG906rVqNel2aLboN2t35rRl+OU5ZHnnumL64jthe+C4Y/zfPV593 b5c/tw/X3/ZPTcA0A80ULHNwTEIxC90AG+bpIcRQE2009HIxjsSKhKY2ckz1UVhIkSN9eCzZIaMW lXROotzAUpLLl2tm0rwQ847NmwByjtp506ceoDSFbiL60mgfpCiVruKJwikgpiWlGrE6iOpIrEK4 8oJ6wqsrrSHFJiL70SwxtBzVMmJb0S0OubrAjqBrEe5EvDX4/rJ7Vy9EvzEIkwT8gYLixYwbO35Q AAA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0134_image049.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0134_image050.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0134_image051.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhWwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABa AAEAgAAAAAAAAAIKjI+pywkPX5uUFgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0134_image052.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0134_image053.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAo AAEAgAAAAAAAAAIFjI+py1gAO1== ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0134_image054.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAb AAEAgAAAAAAAAAIEjI+pWgA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0027.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Henc= e         = ;  
W= 47;t(x) dx =3D Ln( |e(x)| ) + k
Integ= rals of Products of Tangent and Secant
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0027_image055.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMQEqAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAv ASkAgAAAAAAAAALfjI+py+0Po5y02ouz3hyBD4biSJbmiabqyrbuC8fyTNeAYef6zvf+DwyOcMKi 8YhMKoPEpfMJjUp/zan1is1Oq9qu9wueccPksjk8PqvXbGW6DY/La++5/Y4n1fP8PnzvFygIBjho eChViLjIaKTYCBmp8yhZaelCeam5qRfA+QmqkhlKGjlaiop4msoauNoKi/caSxs3W4urdpvLi+bZ C8y3G0xsNVyM/HSczIy03AzN9BtN3fVcjW1znc0ds90NzvIdTn4yXo4ucp7Ovs6O7v5O3kFfb3+P n6+/z99QAAA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0081.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Henc= e         = ;  
W= 47;t(x) dx =3D Ln( |e(x)| ) + k
Integ= rals of Products of Tangent and Secant
Also= by substitution, we get        = ; 
W= 47;e(x) dx =3D Ln( |e(x) + t(x)| ) + = k
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0081_image056.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMQEqAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAv ASkAgAAAAAAAAALfjI+py+0Po5y02ouz3hyBD4biSJbmiabqyrbuC8fyTNeAYef6zvf+DwyOcMKi 8YhMKoPEpfMJjUp/zan1is1Oq9qu9wueccPksjk8PqvXbGW6DY/La++5/Y4n1fP8PnzvFygIBjho eChViLjIaKTYCBmp8yhZaelCeam5qRfA+QmqkhlKGjlaiop4msoauNoKi/caSxs3W4urdpvLi+bZ C8y3G0xsNVyM/HSczIy03AzN9BtN3fVcjW1znc0ds90NzvIdTn4yXo4ucp7Ovs6O7v5O3kFfb3+P n6+/z99QAAA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0081_image057.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhUAEqAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABO ASkAgAAAAAAAAALrjI+py+0Po5y02ouz3rw7AIbiSJbmiabqyrbuC8fyTNf2XRr4zvf+DwwKhzsd 8YhMKpfMZsvojEqn1KoVAL1qt9yu15T9isfkcjBsTqvX7FygDY/Lxei5/Y4n1vP8vj+29yc4SAgS WIiYOHeo2OhYxvgoOXkVSXmJuWSZydn5s+kZKjoDOmp6mlKKusqK9dYKG+smS0urWosreZvLm7jb Cyz4G0ycN1yMLHeczLy23AxN9hxN3TVdjW11nc0dtd0NrvQdTj40Xo7uc57OfrPeDi/zHk//9FqP r3mfz6/n8Q8woMCBBAsaPBiwAAA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0082.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Henc= e         = ;  
W= 47;t(x) dx =3D Ln( |e(x)| ) + k
Integ= rals of Products of Tangent and Secant
Also= by substitution, we get        = ; 
W= 47;e(x) dx =3D Ln( |e(x) + t(x)| ) + = k
= 747; tne dx where n, m > 1, we may convert the integrand to one funct= ion, when poss= ible, and use the reductive formulas.
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0082_image058.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMQEqAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAv ASkAgAAAAAAAAALfjI+py+0Po5y02ouz3hyBD4biSJbmiabqyrbuC8fyTNeAYef6zvf+DwyOcMKi 8YhMKoPEpfMJjUp/zan1is1Oq9qu9wueccPksjk8PqvXbGW6DY/La++5/Y4n1fP8PnzvFygIBjho eChViLjIaKTYCBmp8yhZaelCeam5qRfA+QmqkhlKGjlaiop4msoauNoKi/caSxs3W4urdpvLi+bZ C8y3G0xsNVyM/HSczIy03AzN9BtN3fVcjW1znc0ds90NzvIdTn4yXo4ucp7Ovs6O7v5O3kFfb3+P n6+/z99QAAA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0082_image059.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhUAEqAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABO ASkAgAAAAAAAAALrjI+py+0Po5y02ouz3rw7AIbiSJbmiabqyrbuC8fyTNf2XRr4zvf+DwwKhzsd 8YhMKpfMZsvojEqn1KoVAL1qt9yu15T9isfkcjBsTqvX7FygDY/Lxei5/Y4n1vP8vj+29yc4SAgS WIiYOHeo2OhYxvgoOXkVSXmJuWSZydn5s+kZKjoDOmp6mlKKusqK9dYKG+smS0urWosreZvLm7jb Cyz4G0ycN1yMLHeczLy23AxN9hxN3TVdjW11nc0dtd0NrvQdTj40Xo7uc57OfrPeDi/zHk//9FqP r3mfz6/n8Q8woMCBBAsaPBiwAAA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0082_image060.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0083.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Henc= e         = ;  
W= 47;t(x) dx =3D Ln( |e(x)| ) + k
Integ= rals of Products of Tangent and Secant
Also= by substitution, we get        = ; 
W= 47;e(x) dx =3D Ln( |e(x) + t(x)| ) + = k
= 747; tne dx where n, m > 1, we may convert the integrand to one funct= ion, when poss= ible, and use the reductive formulas.
setting u =3D t(x), or u =3D e(x) depending on the inte= grand.
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0083_image061.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMQEqAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAv ASkAgAAAAAAAAALfjI+py+0Po5y02ouz3hyBD4biSJbmiabqyrbuC8fyTNeAYef6zvf+DwyOcMKi 8YhMKoPEpfMJjUp/zan1is1Oq9qu9wueccPksjk8PqvXbGW6DY/La++5/Y4n1fP8PnzvFygIBjho eChViLjIaKTYCBmp8yhZaelCeam5qRfA+QmqkhlKGjlaiop4msoauNoKi/caSxs3W4urdpvLi+bZ C8y3G0xsNVyM/HSczIy03AzN9BtN3fVcjW1znc0ds90NzvIdTn4yXo4ucp7Ovs6O7v5O3kFfb3+P n6+/z99QAAA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0083_image062.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhUAEqAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABO ASkAgAAAAAAAAALrjI+py+0Po5y02ouz3rw7AIbiSJbmiabqyrbuC8fyTNf2XRr4zvf+DwwKhzsd 8YhMKpfMZsvojEqn1KoVAL1qt9yu15T9isfkcjBsTqvX7FygDY/Lxei5/Y4n1vP8vj+29yc4SAgS WIiYOHeo2OhYxvgoOXkVSXmJuWSZydn5s+kZKjoDOmp6mlKKusqK9dYKG+smS0urWosreZvLm7jb Cyz4G0ycN1yMLHeczLy23AxN9hxN3TVdjW11nc0dtd0NrvQdTj40Xo7uc57OfrPeDi/zHk//9FqP r3mfz6/n8Q8woMCBBAsaPBiwAAA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0083_image063.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0034.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
W= 47; tne dx where m is even, split off e2, convert the rest into t(x), then s= et u =3D t(x) for subs= titution method.   =
Integ= rals of Products of Tangent and Secant
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0084.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Example: Find &#= 8747; t2e4 dx.
W= 47; tne dx where m is even, split off e2, convert the rest into t(x), then s= et u =3D t(x) for subs= titution method.   =
Integ= rals of Products of Tangent and Secant
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0085.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Example: Find &#= 8747; t2e4 dx.
W= 47; tne dx where m is even, split off e2, convert the rest into t(x), then s= et u =3D t(x) for subs= titution method.   =
t2e4 =3D t2e2*e2 &n= bsp;
Integ= rals of Products of Tangent and Secant
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0086.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Example: Find &#= 8747; t2e4 dx.
W= 47; tne dx where m is even, split off e2, convert the rest into t(x), then s= et u =3D t(x) for subs= titution method.   =
t2e4 =3D t2e2*e2 =3D t2(1 + t2)e2
Integ= rals of Products of Tangent and Secant
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0087.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Example: Find &#= 8747; t2e4 dx.
W= 47; tne dx where m is even, split off e2, convert the rest into t(x), then s= et u =3D t(x) for subs= titution method.   =
t2e4 =3D t2e2*e2 =3D t2(1 + t2)e2 =3D(t2 + t4)e2  
Integ= rals of Products of Tangent and Secant
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0088.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Example: Find &#= 8747; t2e4 dx.
W= 47; tne dx where m is even, split off e2, convert the rest into t(x), then s= et u =3D t(x) for subs= titution method.   =
t2e4 =3D t2e2*e2 =3D t2(1 + t2)e2 =3D(t2 + t4)e2  
Integ= rals of Products of Tangent and Secant
Hence ∫= t2e4 dx
=3D ∫= (t2 + t4)e2 dx
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0089.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Example: Find &#= 8747; t2e4 dx.
W= 47; tne dx where m is even, split off e2, convert the rest into t(x), then s= et u =3D t(x) for subs= titution method.   =
t2e4 =3D t2e2*e2 =3D t2(1 + t2)e2 =3D(t2 + t4)e2  
Integ= rals of Products of Tangent and Secant
Hence ∫= t2e4 dx
substitution
Let u =3D t
=3D ∫= (t2 + t4)e2 dx
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0089_image064.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhewBsAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAB5 AGwAgAAAAAAAAAL/jI+py+0Po5wG2Iuz3rz7D4aiWI3miaZqWK7uC8dWK9f2rdH4zrt6DwyCfsKi cRY4KpXEpfPWfEph0ak1Vb1qScmtF9v9irnj8jBsTmey6i+7vX3Dr/L5tG5/4vPLPf/o91cUKBhE WNhziLijuAiF5mjVGCkzSUkFeelkqbnC2QkGKvUpakJaSobKlKkKdNrq8QrLITubw2r7mGuIu1vZ 64sZzFMbXOx7vJucu2zbPPsMG906rVqNel2aLboN2t35rRl+OU5ZHnnumL64jthe+C4Y/zfPV593 b5c/tw/X3/ZPTcA0A80ULHNwTEIxC90AG+bpIcRQE2009HIxjsSKNaY2ckz1UVhIkSN9eCzZIaMW lXROotzAUpLLl2tm0rwQ847NmwByjtp506ceCkSLGj2K1EEBADs= ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0090.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Example: Find &#= 8747; t2e4 dx.
W= 47; tne dx where m is even, split off e2, convert the rest into t(x), then s= et u =3D t(x) for subs= titution method.   =
t2e4 =3D t2e2*e2 =3D t2(1 + t2)e2 =3D(t2 + t4)e2  
Integ= rals of Products of Tangent and Secant
Hence ∫= t2e4 dx
substitution
Let u =3D t
du
dx
 =3D e2
=3D ∫= (t2 + t4)e2 dx
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0090_image065.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAAAAAAIEjI+pUQA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0090_image066.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhewBsAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAB5 AGwAgAAAAAAAAAL/jI+py+0Po5wG2Iuz3rz7D4aiWI3miaZqWK7uC8dWK9f2rdH4zrt6DwyCfsKi cRY4KpXEpfPWfEph0ak1Vb1qScmtF9v9irnj8jBsTmey6i+7vX3Dr/L5tG5/4vPLPf/o91cUKBhE WNhziLijuAiF5mjVGCkzSUkFeelkqbnC2QkGKvUpakJaSobKlKkKdNrq8QrLITubw2r7mGuIu1vZ 64sZzFMbXOx7vJucu2zbPPsMG906rVqNel2aLboN2t35rRl+OU5ZHnnumL64jthe+C4Y/zfPV593 b5c/tw/X3/ZPTcA0A80ULHNwTEIxC90AG+bpIcRQE2009HIxjsSKNaY2ckz1UVhIkSN9eCzZIaMW lXROotzAUpLLl2tm0rwQ847NmwByjtp506ceCkSLGj2K1EEBADs= ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0091.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Example: Find &#= 8747; t2e4 dx.
W= 47; tne dx where m is even, split off e2, convert the rest into t(x), then s= et u =3D t(x) for subs= titution method.   =
t2e4 =3D t2e2*e2 =3D t2(1 + t2)e2 =3D(t2 + t4)e2  
Integ= rals of Products of Tangent and Secant
Hence ∫= t2e4 dx
substitution
Let u =3D t
dx =3D
du
e2 
du
dx
 =3D e2
=3D ∫= (t2 + t4)e2 dx
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0091_image067.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAAAAAAIEjI+pUQA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0091_image068.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegBtAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAB5 AGwAgAAAAAAAAAL/jI+py+0Po5wG2Iuz3rz7D4aiWI3miaZqWK7uC8dWK9f2rdH4zrt6DwyCfsKi cRY4KpXEpfPWfEph0ak1Vb1qScmtF9v9irnj8jBsTmey6i+7vX3Dr/L5tG5/4vPLPf/o91cUKBhE WNhziLijuAiF5mjVGCkzSUkFeelkqbnC2QkGKvUpakJaSobKlKkKdNrq8QrLITubw2r7mGuIu1vZ 64sZzFMbXOx7vJucu2zbPPsMG906rVqNel2aLboN2t35rRl+OU5ZHnnumL64jthe+C4Y/zfPV593 b5c/tw/X3/ZPTcA0A80ULHNwTEIxCSk4+DWMEbCIKha6mUgRhUUvOBvjYMxo6iPIVCOFlTR50ofI lB06anFJZyXLDTAlyZy55ibOCzXv6NwJoOeonzsdGj2KNKnSBAUAADs= ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0092.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Example: Find &#= 8747; t2e4 dx.
W= 47; tne dx where m is even, split off e2, convert the rest into t(x), then s= et u =3D t(x) for subs= titution method.   =
t2e4 =3D t2e2*e2 =3D t2(1 + t2)e2 =3D(t2 + t4)e2  
Integ= rals of Products of Tangent and Secant
Hence ∫= t2e4 dx
substitution
Let u =3D t
dx =3D
du
e2 
du
dx
 =3D e2
 =3D ∫ (u2 + u4)e2
du
e2 
=3D ∫= (t2 + t4)e2 dx
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0092_image069.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAAAAAAIEjI+pUQA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0092_image070.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAa AAEAgAAAAAAAAAIEjI+pWQA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0092_image071.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh2ABtAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAADW AGwAgAAAAAAAAAL/hI+py+0Powm02ouz3rz7D4aBRJbmGY3oyrZu+sbyqc72jR91zs97DwzChETU r4gkHpNMxbIJ9UWniCf1SsNOrdrug+vlgcNkXRk5PofT6he7jX3DV/J5tG4v4fPJPR/i9ycUKMhA WCiGKKXodcg48djl+DgZ6WbZUonJoonYuZkFGira9CloSjqUKoG66tCaB+u6IDtXO1uFC6irxPvq G3TbJqxLfGY8i0ymvMrcCAzkLAndI61lDYp9pY3JvUWdCJ4rfuN9R16ODqkuYw7lzgjPJO/JTl94 X5T/tz/I3m7vX4x+wQReUkfQTsJqBl0sDNeQTsCIEhFSrIju4bCJvxdNaMTx8RjHjiRC2jC5bCRJ VeRQrlG58gvMmA1cDqTJaiZOJyI61Nsp0yLQX0KHGtJpFIDNg0mdIDW61GFTWk+HRs001WnRrOsy ch3n9WvXlmLHiru6rSpQtBi/sjVSVqnanW9HuZ2Ls67HuHr18MVLs2/Jv1u5Cs5Z9jBLw4BjKt6V uPHKx0HFUiZqWTLJyzUJh81cOCvno5FDTx1NtfTnu6abotYKejXj1klfJ7ANUnNH3Dp6+v4NPLjw CgUAADs= ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0094.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Example: Find &#= 8747; t2e4 dx.
W= 47; tne dx where m is even, split off e2, convert the rest into t(x), then s= et u =3D t(x) for subs= titution method.   =
t2e4 =3D t2e2*e2 =3D t2(1 + t2)e2 =3D(t2 + t4)e2  
Integ= rals of Products of Tangent and Secant
Hence ∫= t2e4 dx
substitution
Let u =3D t
dx =3D
du
e2 
du
dx
 =3D e2
 =3D ∫ (u2 + u4)e2
du
e2 
 =3D ∫ (u2 + u4)du
=3D ∫= (t2 + t4)e2 dx
------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0094_image072.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAAAAAAIEjI+pUQA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0094_image073.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAa AAEAgAAAAAAAAAIEjI+pWQA7 ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0094_image074.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh2ABtAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAADW AGwAgAAAAAAAAAL/hI+py+0Powm02ouz3rz7D4aBRJbmGY3oyrZu+sbyqc72jR91zs97DwzChETU r4gkHpNMxbIJ9UWniCf1SsNOrdrug+vlgcNkXRk5PofT6he7jX3DV/J5tG4v4fPJPR/i9ycUKMhA WCiGKKXodcg48djl+DgZ6WbZUonJoonYuZkFGira9CloSjqUKoG66tCaB+u6IDtXO1uFC6irxPvq G3TbJqxLfGY8i0ymvMrcCAzkLAndI61lDYp9pY3JvUWdCJ4rfuN9R16ODqkuYw7lzgjPJO/JTl94 X5T/tz/I3m7vX4x+wQReUkfQTsJqBl0sDNeQTsCIEhFSrIju4bCJvxdNaMTx8RjHjiRC2jC5bCRJ VeRQrlG58gvMmA1cDqTJaiZOJyI61Nsp0yLQX0KHGtJpFIDNg0mdIDW61GFTWk+HRs001WnRrOsy ch3n9WvXlmLHiru6rSpQtBi/sjVSVqnanW9HuZ2Ls67HuHr18MVLs2/Jv1u5Cs5Z9jBLw4BjKt6V uPHKx0HFUiZqWTLJyzUJh81cOCvno5FDTx1NtfTnu6abotYKejXj1klfJ7ANUnNH3Dp6+v4NPLjw CgUAADs= ------=_NextPart_01C97CE7.B7A456A0 Content-Location: file:///C:/EC831234/105IntegralsofProductofTangentandSecant_files/slide0095.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Example: Find &#= 8747; t2e4 dx.
W= 47; tne dx where m is even, split off e2, convert the rest into t(x), then s= et u =3D t(x) for subs= titution method.   =
t2e4 =3D t2e2*e2 =3D t2(1 + t2)e2 =3D(t2 + t4)e2  
Integ= rals of Products of Tangent and Secant
Hence ∫= t2e4 dx
substitution
Let u =3D t
dx =3D
du