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Trig= onometric Substitutions
------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0116.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
integrands containing expressions = of the form
Trig= onometric Substitutions
a2 – x2 ,
a2 + x ,
and<= span style=3D'mso-spacerun:yes'>  x2 – a2.
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integrands containing expressions = of the form
Trig= onometric Substitutions
a2 – x2 ,
a2 + x ,
and<= span style=3D'mso-spacerun:yes'>  x2 – a2.
If we set x to be well chosen trig-functions,
the expressions may be rooted and simplified. 
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integrands containing expressions = of the form
Trig= onometric Substitutions
a2 – x2 ,
a2 + x ,
and<= span style=3D'mso-spacerun:yes'>  x2 – a2.
If we set x to be well chosen trig-functions,
the expressions may be rooted and simplified. 
For<= span style=3D'mso-spacerun:yes'> 
a2 – x2 , if x =3D a*sin(=   )  <= /div>
q
------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0129_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0129_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0129_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0129_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0130.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
integrands containing expressions = of the form
Trig= onometric Substitutions
a2 – x2 ,
a2 + x ,
and<= span style=3D'mso-spacerun:yes'>  x2 – a2.
If we set x to be well chosen trig-functions,
the expressions may be rooted and simplified. 
For<= span style=3D'mso-spacerun:yes'> 
a2 – x2 , if x =3D a*sin(=   )  <= /div>
q
a2 – x2 =3D acos(=   )
q
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integrands containing expressions = of the form
Trig= onometric Substitutions
a2 – x2 ,
a2 + x ,
and<= span style=3D'mso-spacerun:yes'>  x2 – a2.
If we set x to be well chosen trig-functions,
the expressions may be rooted and simplified. 
For<= span style=3D'mso-spacerun:yes'> 
a2 – x2 , if x =3D a*sin(=   )  <= /div>
q
a2 – x2 =3D acos(=   )
q
q
-π/2
π/2
(If
<<= /span>
<<= /span>
)
------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0131_image016.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0131_image017.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0131_image018.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0131_image019.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0131_image020.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0131_image021.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA+ AAEAgAAAAAAAAAIKjI95wO0Pn5qpAAA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0132.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
integrands containing expressions = of the form
Trig= onometric Substitutions
a2 – x2 ,
a2 + x ,
and<= span style=3D'mso-spacerun:yes'>  x2 – a2.
If we set x to be well chosen trig-functions,
the expressions may be rooted and simplified. 
For<= span style=3D'mso-spacerun:yes'> 
a2 – x2 , if x =3D a*sin(=   )  <= /div>
q
a2 – x2 =3D acos(=   )
q
q
-π/2
π/2
(If
<<= /span>
<<= /span>
)
For<= span style=3D'mso-spacerun:yes'> 
a2 + x , if x =3D a*tan(=   )  <= /div>
q
------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0132_image022.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0132_image023.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0132_image024.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0132_image025.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0132_image026.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0132_image027.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA+ AAEAgAAAAAAAAAIKjI95wO0Pn5qpAAA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0132_image028.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0133.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
integrands containing expressions = of the form
Trig= onometric Substitutions
a2 – x2 ,
a2 + x ,
and<= span style=3D'mso-spacerun:yes'>  x2 – a2.
If we set x to be well chosen trig-functions,
the expressions may be rooted and simplified. 
For<= span style=3D'mso-spacerun:yes'> 
a2 – x2 , if x =3D a*sin(=   )  <= /div>
q
a2 – x2 =3D acos(=   )
q
q
-π/2
π/2
(If
<<= /span>
<<= /span>
)
For<= span style=3D'mso-spacerun:yes'> 
a2 + x , if x =3D a*tan(=   )  <= /div>
q
a2 + x =3D asec(=   )
 q
q
-π/2
π/2
(If
<<= /span>
<<= /span>
)
------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0133_image029.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0133_image030.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0133_image031.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0133_image032.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0133_image033.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0133_image034.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQgADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA+ AAEAgAAAAAAAAAIKjI95wO0Pn5qpAAA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0133_image035.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBD AAEAgAAAAAAAAAIHjI+py+0fCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0133_image036.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABE AAEAgAAAAAAAAAIHjI+py+1vCgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0134.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
integrands containing expressions = of the form
Trig= onometric Substitutions
a2 – x2 ,
a2 + x ,
and<= span style=3D'mso-spacerun:yes'>  x2 – a2.
If we set x to be well chosen trig-functions,
the expressions may be rooted and simplified. 
For<= span style=3D'mso-spacerun:yes'> 
a2 – x2 , if x =3D a*sin(=   )  <= /div>
q
a2 – x2 =3D acos(=   )
q
q
-π/2
π/2
(If
<<= /span>
<<= /span>
)
For<= span style=3D'mso-spacerun:yes'> 
a2 + x , if x =3D a*tan(=   )  <= /div>
q
a2 + x =3D asec(=   )
 q
q
-π/2
π/2
(If
<<= /span>
<<= /span>
)
For<= span style=3D'mso-spacerun:yes'> 
x2 – a2 , if x =3D a*sec(=   )  <= /div>
q
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integrands containing expressions = of the form
Trig= onometric Substitutions
a2 – x2 ,
a2 + x ,
and<= span style=3D'mso-spacerun:yes'>  x2 – a2.
If we set x to be well chosen trig-functions,
the expressions may be rooted and simplified. 
For<= span style=3D'mso-spacerun:yes'> 
a2 – x2 , if x =3D a*sin(=   )  <= /div>
q
a2 – x2 =3D acos(=   )
q
q
-π/2
π/2
(If
<<= /span>
<<= /span>
)
For<= span style=3D'mso-spacerun:yes'> 
a2 + x , if x =3D a*tan(=   )  <= /div>
q
a2 + x =3D asec(=   )
 q
q
-π/2
π/2
(If
<<= /span>
<<= /span>
)
For<= span style=3D'mso-spacerun:yes'> 
x2 – a2 , if x =3D a*sec(=   )  <= /div>
q
x2 – a2 =3D atan(=   )
q
q
-π/2
π/2
(If
<<= /span>
<<= /span>
)
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between a, x and    for each substitution= s.
Trig= onometric Substitutions
q
------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0141.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Trig= onometric Substitutions
For<= span style=3D'mso-spacerun:yes'> 
x = =3D a*q
we'v= e  sin(   ) =3D
x
a
q
between a, x and    for each substitution= s.
q
------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0141_image056.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAABAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0142.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Trig= onometric Substitutions
For<= span style=3D'mso-spacerun:yes'> 
x = =3D a*q
q
we'v= e  sin(   ) =3D
x
a
q
x
a
between a, x and    for each substitution= s.
q
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Trig= onometric Substitutions
For<= span style=3D'mso-spacerun:yes'> 
x = =3D a*q
q
we'v= e  sin(   ) =3D
x
a
q
x
a
a2 – x2=  
between a, x and    for each substitution= s.
q
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Trig= onometric Substitutions
For<= span style=3D'mso-spacerun:yes'> 
x = =3D a*q
q
we'v= e  sin(   ) =3D
x
a
q
q
or <= /span>
=3D = sin-1(     )
x
a
x
a
a2 – x2=  
between a, x and    for each substitution= s.
q
------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0144_image062.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhlwBTAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAgCT AFEAgAAAAAAAAAL/hI+py+0Po5y0hVCz3rz7cwHYR5bm6YTGiLbuS7ErTNe2isj2znN4ousJh5Zg johMgoxApbP3Sz2nNCbDSs1qohCs9tv1XsHkmNhSTqfOUrV7SWK/k1zfPC2X5O+7umfPB+P3ARiI MhhnSFdYwajo49j4CCUUOalnmZF5WYS0yQn0uQVqgthHSijaofrIuoq6YTrkOidLRKtm6wkboUvH u1YLPHaHS+U7ZeyEfDwsoiwIzPwFzTMNVn0Dmy3IrT15ndvqTRkYLlxMflvsvKCu+b6LF/9Ldn5J 39nepXVPmt9k3wSAKwg2U+JP2iKBsW4ZtFfpYRlvEvH0qWixCkZxXC8SMlRAy+NHkIc28lFl0lwJ kSPR/EmpKBLLlg8YzaRZEx7OSpJ28sTkc1YvmAqDBZXnjmg7IzePaiLp9ImMplGfPqt6jCrWLRe6 ev0KNqzYsWTLmj2LNq1aswUAADs= ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0144_image063.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAABAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0144_image064.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAW AAEAgAAAAAAAAAIEjI+pUQA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0144_image065.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODdhOwABAHcAACH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACwAAAAAOwABAIAAAAAB AgMCBoSPqcvtWAA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0145.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Trig= onometric Substitutions
For<= span style=3D'mso-spacerun:yes'> 
x = =3D a*q
q
we'v= e  sin(   ) =3D
x
a
q
q
or <= /span>
=3D = sin-1(     )
x
a
x
a
a2 – x2=  
For<= span style=3D'mso-spacerun:yes'> 
x = =3D a*q
we'v= e  tan(   ) =3D
x
a
q
between a, x and    for each substitution= s.
q
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Trig= onometric Substitutions
For<= span style=3D'mso-spacerun:yes'> 
x = =3D a*q
q
we'v= e  sin(   ) =3D
x
a
q
q
or <= /span>
=3D = sin-1(     )
x
a
x
a
a2 – x2=  
For<= span style=3D'mso-spacerun:yes'> 
x = =3D a*q
q
we'v= e  tan(   ) =3D
x
a
q
x
a
between a, x and    for each substitution= s.
q
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Trig= onometric Substitutions
For<= span style=3D'mso-spacerun:yes'> 
x = =3D a*q
q
we'v= e  sin(   ) =3D
x
a
q
q
or <= /span>
=3D = sin-1(     )
x
a
x
a
a2 – x2=  
For<= span style=3D'mso-spacerun:yes'> 
x = =3D a*q
q
we'v= e  tan(   ) =3D
x
a
q
x
a
a2 + x 
between a, x and    for each substitution= s.
q
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Trig= onometric Substitutions
For<= span style=3D'mso-spacerun:yes'> 
x = =3D a*q
q
we'v= e  sin(   ) =3D
x
a
q
q
or <= /span>
=3D = sin-1(     )
x
a
x
a
a2 – x2=  
For<= span style=3D'mso-spacerun:yes'> 
x = =3D a*q
q
we'v= e  tan(   ) =3D
x
a
q
q
or <= /span>
=3D = tan-1(     )
x
a
x
a
a2 + x 
between a, x and    for each substitution= s.
q
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Trig= onometric Substitutions
For<= span style=3D'mso-spacerun:yes'> 
x = =3D a*q
we'v= e  sec(   ) =3D
x
a
q
------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0119_image089.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0149.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Trig= onometric Substitutions
For<= span style=3D'mso-spacerun:yes'> 
x = =3D a*q
q
we'v= e  sec(   ) =3D
x
a
q
x
a
------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0149_image090.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhlwBTAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAgCT AFEAgAAAAAAAAAL/hI+py+0Po5y0hVCz3rz7cwHYR5bm6YTGiLbuO6kgTNc2i+D2znNyousJh5Zg johMgoxApbP3Sz2nsOiDSc1usBatt2ONcL9kRVgyLpfPMbXbnBa/5+xMfJ68t/Hf+pav5ecDOCUI Rqhk6KGHaKL4wdgIKRQpuVW5aEnzeIKpCcGJ4vnJENoySpqD6piambX6abrZagd7Shtj66KLJwuF m8JbBVwKKNx3PEu8klzT7OT79IwUXUhbTTX926p9070tid0X/k1U7nhubpxOzceeR/cOryZOKC92 D428jBaYn53t3ytr/OxAE+glXz1g8hYSY+dw2bmI/MohdOfsIsYXXBQLzjilEdGxjh6blCBZ0uSk lDcghSSFCiXLBZ5kzqRZ6+YQRjZ1FuvnUx2olx7H9AwqpRRRlkaOIgUFZKlOHFJ9jnD6NFdVpBe6 ev0KNqzYsWTLmj2LNq3aswUAADs= ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0149_image091.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0150.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Trig= onometric Substitutions
For<= span style=3D'mso-spacerun:yes'> 
x = =3D a*q
q
we'v= e  sec(   ) =3D
x
a
q
x
a
x2 – a2=  
------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0150_image092.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhlwBTAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAgCT AFEAgAAAAAAAAAL/hI+py+0Po5y0hVCz3rz7cwHYR5bm6YTGiLbuO6kgTNc2i+D2znNyousJh5Zg johMgoxApbP3Sz2nsOiDSc1usBatt2ONcL9kRVgyLpfPMbXbnBa/5+xMfJ68t/Hf+pav5ecDOCUI Rqhk6KGHaKL4wdgIKRQpuVW5aEnzeIKpCcGJ4vnJENoySpqD6piambX6abrZagd7Shtj66KLJwuF m8JbBVwKKNx3PEu8klzT7OT79IwUXUhbTTX926p9070tid0X/k1U7nhubpxOzceeR/cOryZOKC92 D428jBaYn53t3ytr/OxAE+glXz1g8hYSY+dw2bmI/MohdOfsIsYXXBQLzjilEdGxjh6blCBZ0uSk lDcghSSFCiXLBZ5kzqRZ6+YQRjZ1FuvnUx2olx7H9AwqpRRRlkaOIgUFZKlOHFJ9jnD6NFdVpBe6 ev0KNqzYsWTLmj2LNq3aswUAADs= ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0150_image093.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0150_image094.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C97CE5.8A4CBE10 Content-Location: file:///C:/F1764A33/18Trig-Substitutions_files/slide0151.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Trig= onometric Substitutions
For<= span style=3D'mso-spacerun:yes'> 
x = =3D a*q
q
we'v= e  sec(   ) =3D
x
a
q
q
or <= /span>
=3D = sec-1(    )
x
a
x
a
x2 – a2=  
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