MAT 1235 Calculus II Section 7.4 Partial Fractions .

MAT 1235Calculus II

Section 7.4

Partial Fractions

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HW

Please do your HW ASAP. Please actually do your HW.

Partial Fractions

Review PF from pre-calculus Use PF to simplify integrands Break up a complicated rational function

into smaller ones Each of the smaller rational function is

easier to integrate

Preview

Use Partial Fractions to decompose a rational function into a sum of simpler rational functions

dxxQ

xPdx

xQ

xPdx

xQ

xPdx

xQ

xP

)(

)(

)(

)(

)(

)(

)(

)(

2

2

2

2

1

1

Assumption

We assume: deg(P(x))<deg(Q(x))

If this is not the case, we can use long division to rewrite the rational function as

)(

)()(

)(

)(

xQ

xRxS

xQ

xP

1

11

12

3

xxx

x

x

( )

( )

P xdx

Q x

2

1

xdx

x 2

3

3xdx

x x

Assumption

We assume: deg(P(x))<deg(Q(x))

If this is not the case, we can use long division to rewrite the rational function as

)(

)()(

)(

)(

xQ

xRxS

xQ

xP

1

11

12

3

xxx

x

x

( )

( )

P xdx

Q x

Quotient

Remainder

Example 1 2

1

xdx

x

)(

)()(

)(

)(

xQ

xRxS

xQ

xP

Remark

In stead of using the substitution, we can use the following formula

Caxdxax

ln1

Example 2

1

xdx

x

Assumption

We assume: deg(P(x))<deg(Q(x)) Depends on the form of Q(x), we have 3

different cases.

( )

( )

P xdx

Q x

Case I

Q(x) is a product of distinct linear factors

( 1)( 2)

5xd

x xx

Case I

Q(x) is a product of distinct linear factors

kk

k

kk

bxa

A

bxa

A

bxa

A

xQ

xR

bxabxabxaxQ

22

2

11

1

2211

)(

)(

)())(()(

( 1)( 2)

5xd

x xx

( 1)( 2) 2

5

1

x A B

x x x x

Example 3

dxxx

x

)2)(1(

5

: Compare coefficeints

: Plug in numbers

Method I

Method II

( 1)( 2) 2

5

1

x A B

x x x x

Expectation

Make sure you write down the final partial fractions (on the right hand side) before you proceed to evaluate the integral.

To save time…

We will work on the partial fractions only We are not going to actually complete

the integration

Case II

If (axi+bi) is repeated r times

2

1

(( 1) 2)xdx

x

Case II

If (axi+bi) is repeated r times, we use

1 2

2r

ri i i i i i

A A A

a x b a x b a x b

( ) ( )ri iQ x a x b

2 2

1

( 2) 2( 1) 1 ( 1)

C

x x

A B

x x x

2

1

(( 1) 2)xdx

x

Example 4

dx

xx )2()1(

12

Example 4

dxxx )2()1(

12

2 2

2

1

( 1) ( 2) 1 ( 1) 2

1 ( 1)( 2) ( 2) ( 1)

A B C

x x x x x

A x x B x C x

Example 4

dxxx )2()1(

12

2 2

2

2 2

2

1

( 1) ( 2) 1 ( 1) 2

1 ( 1)( 2) ( 2) ( 1)

( 1) ( 2) ( 1) ( 2)

1 ( 1)( 2) ( 2) ( 1)

A B C

x x x x x

A x x B x C x

x x x x

A x x B x C x

Example 4

dxxx )2()1(

12

CBA

xCxBxxA

,,for Solve

)1()2()2)(1(1 2

2 2

1

( 1) ( 2) 1 ( 1) 2

A B C

x x x x x

Case III

If Q(x) has a irreducible factor ax2+bx+c,

2( 1

1

)xdxx

Case III

If Q(x) has a irreducible factor ax2+bx+c,

we use

2

Ax B

ax bx c

2( 1

1

)xdxx 2 2

1

( 1) 1

Ax

x

C

x

B

x x

Example 5

2

1

( 1)dx

x x 2 2

1

( 1) 1

Ax

x

C

x

B

x x