Integration By Partial Fractions ( 29 ( 29 ∫ dx x P x A find; To
X2 T05 06 Partial Fractions
Jul 13, 2015
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Integration By Partial Fractions
( )( )∫ dxxPxA
find; To
Integration By Partial Fractions
( )( )∫ dxxPxA
find; To
( ) ( ) ( ) division a perform ,degdeg If 1 xPxA ≥
Integration By Partial Fractions
( )( )∫ dxxPxA
find; To
( ) ( ) ( ) division a perform ,degdeg If 1 xPxA ≥
( ) ( ) ( ) ( )xPxPxA factorise ,degdeg If 2 <
Integration By Partial Fractions
( )( )∫ dxxPxA
find; To
( ) ( ) ( ) division a perform ,degdeg If 1 xPxA ≥
( ) ( ) ( ) ( )xPxPxA factorise ,degdeg If 2 <
( )ax
Aax
−− write,factor linear for a)
Integration By Partial Fractions
( )( )∫ dxxPxA
find; To
( ) ( ) ( ) division a perform ,degdeg If 1 xPxA ≥
( ) ( ) ( ) ( )xPxPxA factorise ,degdeg If 2 <
( )ax
Aax
−− write,factor linear for a)
( )
( ) ( ) ( ) nax
C
ax
Bax
A
ax
−++
−+
−
−
2
n
write, factorslinear multiplefor b)
Integration By Partial Fractions
( )( )∫ dxxPxA
find; To
( ) ( ) ( ) division a perform ,degdeg If 1 xPxA ≥
( ) ( ) ( ) ( )xPxPxA factorise ,degdeg If 2 <
( )ax
Aax
−− write,factor linear for a)
( )
( ) ( ) ( ) nax
C
ax
Bax
A
ax
−++
−+
−
−
2
n
write, factorslinear multiplefor b)
cbxaxBAx
cbxax++
+++ 22 write, e.g. factors polynomialfor c)
( )∫ +dx
xx
i1
e.g.2
( )∫ +dx
xx
i1
e.g.2
xx
xxx
+
+++2
2 001x
( )∫ +dx
xx
i1
e.g.2
xx
xxx
+
+++2
2 001x
0+− x
1−
( )∫ +dx
xx
i1
e.g.2
xx
xxx
+
+++2
2 001x
0+− x
1−
1−− x
( )∫ +dx
xx
i1
e.g.2
xx
xxx
+
+++2
2 001x
0+− x
1−
1−− x1
( )∫ +dx
xx
i1
e.g.2
xx
xxx
+
+++2
2 001x
∫
++−= dx
xx
11
10+− x
1−
1−− x1
( )∫ +dx
xx
i1
e.g.2
xx
xxx
+
+++2
2 001x
∫
++−= dx
xx
11
1
( ) cxxx +++−= 1log21 2
0+− x
1−
1−− x1
( )∫ +dx
xx
i1
e.g.2
xx
xxx
+
+++2
2 001x
∫
++−= dx
xx
11
1
( ) cxxx +++−= 1log21 2
( )∫ − xxdx
ii 2
3
0+− x
1−
1−− x1
( )∫ +dx
xx
i1
e.g.2
xx
xxx
+
+++2
2 001x
∫
++−= dx
xx
11
1
( ) cxxx +++−= 1log21 2
( )∫ − xxdx
ii 2
3
( )∫ −=
13xxdx
0+− x
1−
1−− x1
( )∫ +dx
xx
i1
e.g.2
xx
xxx
+
+++2
2 001x
∫
++−= dx
xx
11
1
( ) cxxx +++−= 1log21 2
( )∫ − xxdx
ii 2
3
( )∫ −=
13xxdx
0+− x
1−
1−− x1
( )13
1 −=
−+
xxxB
xA
( )∫ +dx
xx
i1
e.g.2
xx
xxx
+
+++2
2 001x
∫
++−= dx
xx
11
1
( ) cxxx +++−= 1log21 2
( )∫ − xxdx
ii 2
3
( )∫ −=
13xxdx
0+− x
1−
1−− x1
( )13
1 −=
−+
xxxB
xA
( ) 31 =+− BxxA
( )∫ +dx
xx
i1
e.g.2
xx
xxx
+
+++2
2 001x
∫
++−= dx
xx
11
1
( ) cxxx +++−= 1log21 2
( )∫ − xxdx
ii 2
3
( )∫ −=
13xxdx
0+− x
1−
1−− x1
( )13
1 −=
−+
xxxB
xA
( ) 31 =+− BxxA
3
3
0
−==−=
A
A
x
( )∫ +dx
xx
i1
e.g.2
xx
xxx
+
+++2
2 001x
∫
++−= dx
xx
11
1
( ) cxxx +++−= 1log21 2
( )∫ − xxdx
ii 2
3
( )∫ −=
13xxdx
0+− x
1−
1−− x1
( )13
1 −=
−+
xxxB
xA
( ) 31 =+− BxxA
3
3
0
−==−=
A
A
x
3
1
==
B
x
( )∫ +dx
xx
i1
e.g.2
xx
xxx
+
+++2
2 001x
∫
++−= dx
xx
11
1
( ) cxxx +++−= 1log21 2
( )∫ − xxdx
ii 2
3
( )∫ −=
13xxdx
( )∫
−+−= dx
xx 133
0+− x
1−
1−− x1
( )13
1 −=
−+
xxxB
xA
( ) 31 =+− BxxA
3
3
0
−==−=
A
A
x
3
1
==
B
x
( )∫ +dx
xx
i1
e.g.2
xx
xxx
+
+++2
2 001x
∫
++−= dx
xx
11
1
( ) cxxx +++−= 1log21 2
( )∫ − xxdx
ii 2
3
( )∫ −=
13xxdx
( )∫
−+−= dx
xx 133
( ) cxx +−+−= 1log3log3
0+− x
1−
1−− x1
( )13
1 −=
−+
xxxB
xA
( ) 31 =+− BxxA
3
3
0
−==−=
A
A
x
3
1
==
B
x
( )∫ +dx
xx
i1
e.g.2
xx
xxx
+
+++2
2 001x
∫
++−= dx
xx
11
1
( ) cxxx +++−= 1log21 2
( )∫ − xxdx
ii 2
3
( )∫ −=
13xxdx
( )∫
−+−= dx
xx 133
( ) cxx +−+−= 1log3log3
0+− x
1−
1−− x1
( )13
1 −=
−+
xxxB
xA
( ) 31 =+− BxxA
3
3
0
−==−=
A
A
x
3
1
==
B
x
cx
x +
−= 1
log3
( )∫ −−+
dxxx
xiii
1035
2
( )∫ −−+
dxxx
xiii
1035
2
( ) ( )∫ +−+= dx
xxx
255
( )∫ −−+
dxxx
xiii
1035
2
( ) ( )∫ +−+= dx
xxx
255
( ) ( ) ( ) ( )( ) ( ) 552
255
25
+=−+++−
+=+
+−
xxBxA
xxx
xB
xA
( )∫ −−+
dxxx
xiii
1035
2
( ) ( )∫ +−+= dx
xxx
255
( ) ( ) ( ) ( )( ) ( ) 552
255
25
+=−+++−
+=+
+−
xxBxA
xxx
xB
xA
73
37
2
−=
=−−=
B
B
x
( )∫ −−+
dxxx
xiii
1035
2
( ) ( )∫ +−+= dx
xxx
255
( ) ( ) ( ) ( )( ) ( ) 552
255
25
+=−+++−
+=+
+−
xxBxA
xxx
xB
xA
73
37
2
−=
=−−=
B
B
x
710
107
5
=
==
A
A
x
( )∫ −−+
dxxx
xiii
1035
2
( ) ( )∫ +−+= dx
xxx
255
( ) ( ) ( ) ( )( ) ( ) 552
255
25
+=−+++−
+=+
+−
xxBxA
xxx
xB
xA
73
37
2
−=
=−−=
B
B
x
710
107
5
=
==
A
A
x
( ) ( )∫
+−
−= dx
xx 273
5710
( )∫ −−+
dxxx
xiii
1035
2
( ) ( )∫ +−+= dx
xxx
255
( ) ( ) ( ) ( )( ) ( ) 552
255
25
+=−+++−
+=+
+−
xxBxA
xxx
xB
xA
73
37
2
−=
=−−=
B
B
x
710
107
5
=
==
A
A
x
( ) ( )∫
+−
−= dx
xx 273
5710
( ) ( ) cxx ++−−= 2log73
5log7
10
( )∫ −−+
dxxx
xiii
1035
2
( ) ( )∫ +−+= dx
xxx
255
( ) ( ) ( ) ( )( ) ( ) 552
255
25
+=−+++−
+=+
+−
xxBxA
xxx
xB
xA
73
37
2
−=
=−−=
B
B
x
710
107
5
=
==
A
A
x
( ) ( )∫
+−
−= dx
xx 273
5710
( ) ( ) cxx ++−−= 2log73
5log7
10
( )∫ + xxdx
iv 3
( )∫ −−+
dxxx
xiii
1035
2
( ) ( )∫ +−+= dx
xxx
255
( ) ( ) ( ) ( )( ) ( ) 552
255
25
+=−+++−
+=+
+−
xxBxA
xxx
xB
xA
73
37
2
−=
=−−=
B
B
x
710
107
5
=
==
A
A
x
( ) ( )∫
+−
−= dx
xx 273
5710
( ) ( ) cxx ++−−= 2log73
5log7
10
( )∫ + xxdx
iv 3
( )∫ +=
12xxdx
( )∫ −−+
dxxx
xiii
1035
2
( ) ( )∫ +−+= dx
xxx
255
( ) ( ) ( ) ( )( ) ( ) 552
255
25
+=−+++−
+=+
+−
xxBxA
xxx
xB
xA
73
37
2
−=
=−−=
B
B
x
710
107
5
=
==
A
A
x
( ) ( )∫
+−
−= dx
xx 273
5710
( ) ( ) cxx ++−−= 2log73
5log7
10
( )∫ + xxdx
iv 3
( )∫ +=
12xxdx
( )( ) ( ) 11
11
12
22
=++++
=+++
xCBxxA
xxxCBx
xA
( )∫ −−+
dxxx
xiii
1035
2
( ) ( )∫ +−+= dx
xxx
255
( ) ( ) ( ) ( )( ) ( ) 552
255
25
+=−+++−
+=+
+−
xxBxA
xxx
xB
xA
73
37
2
−=
=−−=
B
B
x
710
107
5
=
==
A
A
x
( ) ( )∫
+−
−= dx
xx 273
5710
( ) ( ) cxx ++−−= 2log73
5log7
10
( )∫ + xxdx
iv 3
( )∫ +=
12xxdx
( )( ) ( ) 11
11
12
22
=++++
=+++
xCBxxA
xxxCBx
xA
1
0
==
A
x
( )∫ −−+
dxxx
xiii
1035
2
( ) ( )∫ +−+= dx
xxx
255
( ) ( ) ( ) ( )( ) ( ) 552
255
25
+=−+++−
+=+
+−
xxBxA
xxx
xB
xA
73
37
2
−=
=−−=
B
B
x
710
107
5
=
==
A
A
x
( ) ( )∫
+−
−= dx
xx 273
5710
( ) ( ) cxx ++−−= 2log73
5log7
10
( )∫ + xxdx
iv 3
( )∫ +=
12xxdx
( )( ) ( ) 11
11
12
22
=++++
=+++
xCBxxA
xxxCBx
xA
1
0
==
A
x
1=+−=
CiB
ix
( )∫ −−+
dxxx
xiii
1035
2
( ) ( )∫ +−+= dx
xxx
255
( ) ( ) ( ) ( )( ) ( ) 552
255
25
+=−+++−
+=+
+−
xxBxA
xxx
xB
xA
73
37
2
−=
=−−=
B
B
x
710
107
5
=
==
A
A
x
( ) ( )∫
+−
−= dx
xx 273
5710
( ) ( ) cxx ++−−= 2log73
5log7
10
( )∫ + xxdx
iv 3
( )∫ +=
12xxdx
( )( ) ( ) 11
11
12
22
=++++
=+++
xCBxxA
xxxCBx
xA
1
0
==
A
x
1=+−=
CiB
ix
0 1 =−= CB
( )∫ −−+
dxxx
xiii
1035
2
( ) ( )∫ +−+= dx
xxx
255
( ) ( ) ( ) ( )( ) ( ) 552
255
25
+=−+++−
+=+
+−
xxBxA
xxx
xB
xA
73
37
2
−=
=−−=
B
B
x
710
107
5
=
==
A
A
x
( ) ( )∫
+−
−= dx
xx 273
5710
( ) ( ) cxx ++−−= 2log73
5log7
10
( )∫ + xxdx
iv 3
( )∫ +=
12xxdx
( )( ) ( ) 11
11
12
22
=++++
=+++
xCBxxA
xxxCBx
xA
1
0
==
A
x
1=+−=
CiB
ix
0 1 =−= CB∫
+−= dx
xx
x 11
2
( )∫ −−+
dxxx
xiii
1035
2
( ) ( )∫ +−+= dx
xxx
255
( ) ( ) ( ) ( )( ) ( ) 552
255
25
+=−+++−
+=+
+−
xxBxA
xxx
xB
xA
73
37
2
−=
=−−=
B
B
x
710
107
5
=
==
A
A
x
( ) ( )∫
+−
−= dx
xx 273
5710
( ) ( ) cxx ++−−= 2log73
5log7
10
( )∫ + xxdx
iv 3
( )∫ +=
12xxdx
( )( ) ( ) 11
11
12
22
=++++
=+++
xCBxxA
xxxCBx
xA
1
0
==
A
x
1=+−=
CiB
ix
0 1 =−= CB∫
+−= dx
xx
x 11
2
( ) cxx ++−= 1log21
log 2
( )( ) ( )∫ ++ 11 22 xx
xdxv
( )( ) ( )∫ ++ 11 22 xx
xdxv
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) xxDCxxBxxA
xx
xx
DCx
x
Bx
A
=+++++++
++=
+++
++
+222
2222
1111
11111
( )( ) ( )∫ ++ 11 22 xx
xdxv
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) xxDCxxBxxA
xx
xx
DCx
x
Bx
A
=+++++++
++=
+++
++
+222
2222
1111
11111
21
12
1
−=
−=−=
B
B
x
( )( ) ( )∫ ++ 11 22 xx
xdxv
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) xxDCxxBxxA
xx
xx
DCx
x
Bx
A
=+++++++
++=
+++
++
+222
2222
1111
11111
21
12
1
−=
−=−=
B
B
x
iDiC
ix
=+−=
22
( )( ) ( )∫ ++ 11 22 xx
xdxv
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) xxDCxxBxxA
xx
xx
DCx
x
Bx
A
=+++++++
++=
+++
++
+222
2222
1111
11111
21
12
1
−=
−=−=
B
B
x
iDiC
ix
=+−=
22
21
0 == DC
( )( ) ( )∫ ++ 11 22 xx
xdxv
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) xxDCxxBxxA
xx
xx
DCx
x
Bx
A
=+++++++
++=
+++
++
+222
2222
1111
11111
21
12
1
−=
−=−=
B
B
x
iDiC
ix
=+−=
22
21
0 == DC
0
021
21
2
02
0
=
=+−
=++=
A
A
DBA
x
( )( ) ( )∫ ++ 11 22 xx
xdxv
( ) ( )∫
+
++
−= dxxx 12
1
12
122
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) xxDCxxBxxA
xx
xx
DCx
x
Bx
A
=+++++++
++=
+++
++
+222
2222
1111
11111
21
12
1
−=
−=−=
B
B
x
iDiC
ix
=+−=
22
21
0 == DC
0
021
21
2
02
0
=
=+−
=++=
A
A
DBA
x
( )( ) ( )∫ ++ 11 22 xx
xdxv
( ) ( )∫
+
++
−= dxxx 12
1
12
122
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) xxDCxxBxxA
xx
xx
DCx
x
Bx
A
=+++++++
++=
+++
++
+222
2222
1111
11111
21
12
1
−=
−=−=
B
B
x
iDiC
ix
=+−=
22
( ) ( )∫
+++−= − dx
xx
11
121
2
2 21
0 == DC
0
021
21
2
02
0
=
=+−
=++=
A
A
DBA
x
( )( ) ( )∫ ++ 11 22 xx
xdxv
( ) ( )∫
+
++
−= dxxx 12
1
12
122
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) xxDCxxBxxA
xx
xx
DCx
x
Bx
A
=+++++++
++=
+++
++
+222
2222
1111
11111
21
12
1
−=
−=−=
B
B
x
iDiC
ix
=+−=
22
( ) ( )∫
+++−= − dx
xx
11
121
2
2
( )cx
x +
+−+−= −
−1
1
tan11
21
21
0 == DC
0
021
21
2
02
0
=
=+−
=++=
A
A
DBA
x
( )( ) ( )∫ ++ 11 22 xx
xdxv
( ) ( )∫
+
++
−= dxxx 12
1
12
122
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) xxDCxxBxxA
xx
xx
DCx
x
Bx
A
=+++++++
++=
+++
++
+222
2222
1111
11111
21
12
1
−=
−=−=
B
B
x
iDiC
ix
=+−=
22
( ) ( )∫
+++−= − dx
xx
11
121
2
2
( )cx
x +
+−+−= −
−1
1
tan11
21
cxx
+
+
+= −1tan
11
21
21
0 == DC
0
021
21
2
02
0
=
=+−
=++=
A
A
DBA
x
( )( ) ( )∫ ++ 11 22 xx
xdxv
( ) ( )∫
+
++
−= dxxx 12
1
12
122
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) xxDCxxBxxA
xx
xx
DCx
x
Bx
A
=+++++++
++=
+++
++
+222
2222
1111
11111
21
12
1
−=
−=−=
B
B
x
iDiC
ix
=+−=
22
( ) ( )∫
+++−= − dx
xx
11
121
2
2
( )cx
x +
+−+−= −
−1
1
tan11
21
cxx
+
+
+= −1tan
11
21
21
0 == DC
0
021
21
2
02
0
=
=+−
=++=
A
A
DBA
x
Exercise 2G; 1, 3, 5, 7 to 21
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