Section 7.4: Partial Fractions

3

1

x xdx

x

31x x x 3 2x x

2x

2x

2x2x x

x 2

2 2x

2

1x

2

2 22

1x dx

xx

3 21 1

2 2ln | 1|3 2x x x x C

Long Division works if degree of top ≥ degree of bottom

x

3 2

( 2)( 1)

xdx

x x

( 1) ( 2)

( 2)( 1)

x x

x

B

x

A

Partial Fractions

( 1) ( 2) 3 2x x xA B

2 1

A Bdx

x x

3 2

( 2)( 1)

x

x x

3A B

( ) 2 3 2x xA B A B

2 2A B

3 2

( 2)( 1)

xdx

x x

Partial Fractions

2 1

A Bdx

x x

8

3A

3

2 2

A B

A B

3 1B

1

3B

33

1A

3 2

( 2)( 1)

xdx

x x

8 / 3 1/ 3

2 1dx

x x

8 1ln | 2 | ln | 1|3 3

x x C

Heaviside Method

1x

The Equation is true no matter what x is! So try a few specific x’s to “strike” factors.

( 1) ( 2) 3 2x x xA B

( 1) (1 1 (1)2) 3 2A B

3 1B

1

3B

Heaviside Method

2x

The Equation is true no matter what x is! So try a few specific x’s to “strike” factors.

( 1) ( 2) 3 2x x xA B

2 2( 1) ( 2 3( 2) 2)A B

3 8A

8

3A

2

3

2 4

4

x xdx

x x

x2 +4 is irreducible quadratic

2 4x

BxA C

x

Note:Heaviside isn’t much use here!x=0 will get rid of B & C but nothing will get rid of x2+4

2

2

2 4

( 4)

x xdx

x x

2 2 2( 4) 2 4x x xA CxB x

1C

2

2

2 4

( 4)

x xdx

x x

2 2 2( 4) 2 4x x xA C xB x

2 4

A B Cxdx

x x

1C

1A2A B

2 24 2 4( ) x x x xC AA B

4 4A

1B

ln | |x

2

2

2 4

( 4)

x xdx

x x

2 4

A B Cxdx

x x

2

1 1 1

4

xdx

x x

2 2

1 1

4 4

xdx

x x x

21ln | 4 |

2x 11

tan2 2

xC

General Rules:Linear terms get constants, like AQuadratic terms get linear terms, like Bx+CMultiple terms get repeated

5 3

2 3 2 2

7 5

( 2)( 7)( 4) ( 1)

x x

x x x x x

2

A

x 2 2( 1)

x

x

I J

2 7

x

x

C

x

B

4

D

x

2( 4)

E

x

3( 4)

F

x

2 1

G Hx

x

3

4

1

xdx

x x

2 4C

2

4

( 1)( 1)

xdx

x x

21 1 ( 1)dx

x x x

A B C

2( 1) ( 1)( 1) ( 1) 4A Bx x x xC x

1x 2C

1x 4 4A 1A

3

4

1

xdx

x x

0A B C

2

4

( 1)( 1)

x

xdx

x

21 1 ( 1)dx

x x x

A B C

2( 1) ( 1)( 1) ( 1) 4A Bx x x xC x

0x

1B

2C

1A

01 2B

3

4

1

xdx

x x

2

1 1 2

1 1 ( 1)dx

x x x

2

4

( 1)( 1)

x

xdx

x

21 1 ( 1)dx

x x x

A B C

1( 1)ln | 1| ln | 1| 2

1

xx x C

1A

1B

2C

2

1

2 3

xdx

x x

Irreducible. Complete Square.21

1

( ) 2dx

x

x

1u

d x

x

du

2

1

2du

u

x

2 2

2

2 2

udu du

u u

2

( 11)

2du

u

u

1x u

2

1

2 3

xdx

x x

2

1

( 1) 2

xdx

x

2 2

2

2 2

udu du

u u

2 11 2ln | 2 | tan

2 2 2

uu C

2 11 2 1ln | ( 1) 2 | tan

2 2 2

xx C