View
242
Download
0
Category
Tags:
Preview:
Citation preview
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
Example
Find dxx
1
Example
Find dx
x
1
1
Example
Find dx
ax
1
Example
Find dx
x
x
3
2Example
Find
dxx
xx
3
42
Rational function:)(
)()(xq
xpxf
2
1
2
1
xx 4
42 x
dx
xx 2
1
2
1dx
x 4
42
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
1
1
1
12
xx )1)(1( 2
2
xx
xx
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
dxxx
xx
)1)(1( 2
2
dx
xdx
x 1
1
1
12
1
1
1
12
xx )1)(1( 2
2
xx
xx
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
2
1
2
1
xx 4
42 x
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
4
42 x 22
x
B
x
A
Multiply by
)2)(2(
4
xx
)2)(2( xx
)2()2(4 xBxA
Match coeff subsitute
)22()(4 BAxBA
BA0
BA 224
1 subsitute
A44
2
)2()2(4 xBxA
2x
2x B44
3 The Heaviside “Cover-up”
22
x
B
x
A)2)(2(
4
xx
Example
2x
)22(
4
A
22
x
B
x
A)2)(2(
4
xx
2x
)22(
4
B
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
16483
145 xxx )13)(4)(4(
122
xxx
linear factorquadratic
factor quadratic factor
irreduciblereducible
16483
145 xxx )13)(2)(2)(4(
12
xxxx all factors are irreducible
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
)deg()deg( if qp dxxq
xp )(
)( Use long division1
)32)(1()( xxxq
Factor q(x) as linear factors or irreducible quadratic 3)3)(4()( 2 xxxq
5
1
2
12
x
x
x
Express p(x)/q(x) as a sum of partial fraction4 ii cbxax
BAx
bax
A
)(or
)( 2
q(x)= product of linear factor
All distinct Some repeated
q(x)= product of quadratic (irred)
All distinct repeated
case1case2 case3 case4
Check if we can use subsitution2 15
522
xx
x
)5)(3)(2(
1
xxx 2)3)(2(
1
xx )1)(4(
122 xx 2232 )1()4(
1
xx
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
Example
dx
xxx
xxI
23
1223
2
Example
Find dx
x
4
12
q(x)= product of linear factor
All distinct Some repeated
case1case2
)1)(2(
12
23
12 2
23
2
xxx
xx
xxx
xx12
xxxA B C
)1)(2(
1
A)1)(2(
7B
)1)(1(
2
C
dxx
dxx
dxx
I 1
1
2 2
7
2
1
Cxxx 1ln2ln2
7ln
2
1
Cover-up
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
Example
dxxxx
xxxI
1
14223
24
Example
32 )1()1(
5
xx
x
q(x)= product of linear factor
All distinct Some repeated
case1case2
2)1(1
xx 32 )1()1(1
xxx
A C EDB
Remark: only use subsitue method or match coeff to find the constants A,B,C,D,E.
1
4)1(
1
1422323
24
xxx
xx
xxx
xxxLong Division:
Factor: 223 )1)(1(
4
1
4
xx
x
xxx
x
22 )1(11)1)(1(
4
x
C
x
B
x
A
xx
xPartialFraction:
Multiply: )1()1)(1()1(4 2 xCxxBxAx
subsitute: Cx 24 1
Ax 44 1
CBAx 0 0
dxxxx
xI2)1(
2
1
1
1
11Integrate:
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
Example
)3()1()1(
132 xxx
q(x)= product of linear factor
All distinct Some repeated
case1case2
2)1(1
xx 32 )1()1(1
xxx
A C EDB
)3( x
F
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
q(x)= product of quadratic (irred)
All distinct repeated
case3 case4
Example
)4)(1)(2( 22 xxx
x
2x 41 22
xx
EDx CBx A
Example
222 )4)(1( xx
x
12 xBAx FEx
222 )4(4
xxDCx
Expand by partial fraction (DONOT EVALUATE )
Expand by partial fraction (DONOT EVALUATE )
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
Expand by partial
Find the constants
Evaluate the integral
Recommended