integrating rational functions by the method of partial fraction
Post on 19-Dec-2015
238 views
TRANSCRIPT
![Page 1: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/1.jpg)
Integrating Rational Functions by the Method of Partial Fraction
![Page 2: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/2.jpg)
Examples I
When the power of the polynomial of the numerator is less than that
of the denominator
![Page 3: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/3.jpg)
Example 1
dxxxx
x
)12)(1(
4822
![Page 4: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/4.jpg)
)12)(1(
48)(
22
xxx
xxr
Let
)12)(1(
)()2()2()(
)12)(1(
)12)(()1()1)(1(
)12)(1(
)1)(()1()1)(1(
1)1(1)(
)1)(1(
)12)(1(
22
23
22
222
22
222
22
22
22
xxx
dbaxdcaxdcbaxca
xxx
xxdcxxbxxa
xxx
xdcxxbxxa
x
dcx
x
b
x
axr
xx
xxx
The denominator
![Page 5: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/5.jpg)
8
42
02
0
,
)()2()2()(
4823
dba
dca
dcba
ca
getWe
dbaxdcaxdcbaxca
numeratornewtheandxnumeratororiginaltheComparing
![Page 6: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/6.jpg)
246,4)4(
482022)6(
:),2(
682
:),4(
242
42
:),3(
)1(
)4(8
)3(42
)2(02
)1(0
bc
aaaaa
getweinthatngSubstituti
abba
getweinthatngSubstituti
dd
daa
getweinthatngSubstituti
acFrom
dba
dca
dcba
ca
![Page 7: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/7.jpg)
cxxx
x
dxx
x
xx
dxxr
Thus
x
x
xx
x
dcx
x
b
x
axr
d
c
b
a
getweequationslinearofsystemthisSolving
arctan2)1ln(21
)1(21ln4
]1
24
)1(
2
1
4[
)(
,
1
24
)1(
2
1
4
1)1(1)(
2
4
2
4
,,
21
22
22
22
![Page 8: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/8.jpg)
Example 2
xx
dx2
![Page 9: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/9.jpg)
)1(
11)(
,
11
,,
10
,,
)1(
)(
)1(
)1(
)1()1(
1
)1(
11)(
2
xxxr
Thus
banda
getweequationslinearofsystemthisSolving
aandba
getwenumeratorstwotheComparing
xx
axba
xx
bxxa
x
b
x
a
xx
Let
xxxxxr
![Page 10: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/10.jpg)
cx
x
cxx
dxxx
dxxr
Thus
1ln
1lnln
])1(
11[
)(
,
![Page 11: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/11.jpg)
Example 3
dxx
xx
22
2
)1(
12
![Page 12: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/12.jpg)
01,2
1,2,1,0
,
)()(
)()1)((
)1()1(
)1(
12)(
)1(
12
23
2
222
22
2
22
2
bdc
dbcaba
SolvingandnumeratorsComparing
dbxcabxax
dcxxbax
numeratornewThe
x
dcx
x
bax
x
xxxr
dxx
xx
![Page 13: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/13.jpg)
cx
x
cx
x
dxx
x
x
dxxr
1
1arctan
1
)1(arctan
])1(
2
)1(
1[
)(
2
12
222
![Page 14: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/14.jpg)
Examples II
When the power of the polynomial of the numerator is equal or greater than
that of the denominator
![Page 15: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/15.jpg)
Example 1
dxx
x
1
22
![Page 16: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/16.jpg)
cxxx
dxx
xdxx
x
Thus
xx
x
x
x
x
methodAlternativ
xx
x
xxr
getweDividing
dxx
x
1ln32
]1
31[
1
2
,
1
31
1
3)1(
1
2
:
1
31
1
2)(
,,1
2
2
2
22
2
2
![Page 17: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/17.jpg)
Notice
We can use the same method used in this example as an alternative way to write the given rational function as a sum of simpler rational functions (partial fractions).
Going back to examples. Notice the following:
![Page 18: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/18.jpg)
)1(
11
)1()1(
1
)1(
)1(
)1(
1
1
2
2
xx
xx
x
xx
x
xx
xx
xx
xx
Example
![Page 19: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/19.jpg)
222
22
2
22
2
)1(
2
1
1
)1(
2)1(
)1(
12
3
x
x
x
x
xx
x
xx
Example
![Page 20: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/20.jpg)
Question( Use two methods)
12x
dx
![Page 21: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/21.jpg)
Example 2
12
3
x
dxx
![Page 22: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/22.jpg)
1ln2
1
2)(
1
1
)(
1)(
1
22
2
2
3
2
3
2
3
xx
dxxr
x
xx
x
xxx
x
xxr
x
dxx
![Page 23: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/23.jpg)
Examples III
Sometimes it is easier to find the constants not by solving a system of linear equations but rather by substituting a different appropriate value for x, in each of these equationsز
![Page 24: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/24.jpg)
Example 1
dxxxxxxx
x
)5)(4)(3)(2)(1(
12015
![Page 25: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/25.jpg)
54321
)5)(4)(3)(2)(1(
12015)(
)5)(4)(3)(2)(1(
12015
543210
x
a
x
a
x
a
x
a
x
a
x
a
xxxxxx
xxr
dxxxxxxx
x
![Page 26: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/26.jpg)
)4)(3)(2)(1(
)5)(3)(2)(1(
)5)(4)(2)(1(
)5)(4)(3)(1(
)5)(4)(3)(2(
)5)(4)(3)(2)(1(
5
4
3
2
1
0
xxxxxa
xxxxxa
xxxxxa
xxxxxa
xxxxxa
xxxxxa
numeratornewThe
![Page 27: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/27.jpg)
1)5)(4)(3)(2(
120
120)5)(4)(3)(2)(1(
,
120120)0(15
)40)(30)(20)(10)(0(
)50)(30)(20)(10)(0(
)50)(40)(20)(10)(0(
)50)(40)(30)(10)(0(
)50)(40)(30)(20)(0(
)50)(40)(3)(20)(10(
0,
0
0
5
4
3
2
1
0
a
a
Thus
numeratororiginalThe
a
a
a
a
a
xa
numeratornewThe
xSubstitute
![Page 28: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/28.jpg)
8
35
)4)(3)(2(
105
105)4)(3)(2)(1)(1(
,
105120)1(15
)51)(41)(31)(21)(1(
1
1
1
1
a
a
Thus
numeratororiginalThe
a
numeratornewThe
xSubstitute
![Page 29: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/29.jpg)
4
25
)2)(1)(1)(2)(3(
75
120)3(15)2)(1)(1)(2)(3(
,,3
2
15
)3)(2)(1)(1)(2(
90
120)2(15)3)(2)(1)(1)(2(
,,2
3
3
2
2
a
a
getwexSubstitute
a
a
getwexSubstitute
![Page 30: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/30.jpg)
8
3
)1)(2)(3)(4)(5(
45
120)5(15)1)(2)(3)(4)(5(
,,5
2
5
)1)(1)(2)(3)(4(
60
120)4(15)1)(1)(2)(3)(4(
,,4
5
5
4
4
a
a
getwexSubstitute
a
a
getwexSubstitute
![Page 31: Integrating Rational Functions by the Method of Partial Fraction](https://reader036.vdocuments.us/reader036/viewer/2022062320/56649d395503460f94a12987/html5/thumbnails/31.jpg)
cxx
xxxx
dxxxxxxx
dxxr
Therfore
5ln8
34ln
2
5
3ln2
252ln
2
151ln
8
35ln
]583
425
3225
2215
1835
1[
)(
,