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