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Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS Example Find dx x 1 Example Find dx x 1 1 Example Find dx a x 1 Example Find dx x x 3 2 Example Find dx x x x 3 4 2 Rational function : ) ( ) ( ) ( x q x p x f

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Page 1: Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS Example Find Example Find Example Find Example Find Example Find Rational function:

Sec 7.4: 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

Page 2: Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS Example Find Example Find Example Find Example Find Example Find Rational function:

2

1

2

1

xx 4

42 x

2

1

2

1

xx4

42 x

dx

xx 2

1

2

1dx

x 4

42

Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

Page 3: Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS Example Find Example Find Example Find Example Find Example Find Rational function:

Sec 7.4: 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 fraction4ii cbxax

BAx

bax

A

)(or

)( 2

q(x)= product of linear factor

All distinct Some repeated

q(x)= product of quadratic (irred)

All distinct repeatedcase1

case2 case3 case4

Check if we can use subsitution2 15

522

xx

x

Page 4: Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS Example Find Example Find Example Find Example Find Example Find Rational function:

Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

Example Find

dxxxx

xx

232

1223

2

Example

Find dx

x

4

12

q(x)= product of linear factor

All distinct Some repeated

case1case2

Page 5: Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS Example Find Example Find Example Find Example Find Example Find Rational function:

Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

Example

Find dx

xxx

xxx

1

14223

24

Example

32 )1(

1

xx

x

q(x)= product of linear factor

All distinct Some repeated

case1case2

322 )1()1(1

x

E

x

D

x

C

x

B

x

A

Page 6: Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS Example Find Example Find Example Find Example Find Example Find Rational function:

Sec 7.4: 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

412 22

x

EDx

x

CBx

x

A

Example

Find dx

xx

xx

4

423

2

Page 7: Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS Example Find Example Find Example Find Example Find Example Find Rational function:

Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

q(x)= product of quadratic (irred)

All distinct repeated

case3 case4

Example

222 )4)(1( xx

x

)4(41 222

x

FEx

x

DCx

x

BAx

Example

Express only

dxxxxxx

xx

)1)(1)(1(

1322

23

Example

Find

dxxx

xxx

)1(

2122

32

Page 8: Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS Example Find Example Find Example Find Example Find Example Find Rational function:

Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

RATIONALIZING SUBSTITUTIONS

Some nonrational functions can be changed into rational functions

n xg )(contains )(xf

Example

Find

dxx

x

4

n xgu )( :use

Page 9: Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS Example Find Example Find Example Find Example Find Example Find Rational function:

Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

rational function of sin and cos

2

2

1

1cos

t

tx

Use the following to convert it into rational function121

2sin

t

tx

dt

tdx

21

2

xt x )

2tan(

Evaluate the integral then use:2


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