8.5 Partial Fractions

2

5 3

2 3

xdx

x x

This would be a lot easier if we could

re-write it as two separate terms.

5 3

3 1

x

x x

3 1

A B

x x

1

These are called non-repeating linear factors.

You may already know a short-cut for this type of problem. We will get to that in a few minutes.

2

5 3

2 3

xdx

x x

This would be a lot easier if we could

re-write it as two separate terms.

5 3

3 1

x

x x

3 1

A B

x x

Multiply by the common denominator.

5 3 1 3x A x B x

5 3 3x Ax A Bx B Set like-terms equal to each other.

5x Ax Bx 3 3A B

5 A B 3 3A B Solve two equations with two unknowns.

1

2

5 3

2 3

xdx

x x

5 3

3 1

x

x x

3 1

A B

x x

5 3 1 3x A x B x

5 3 3x Ax A Bx B

5x Ax Bx 3 3A B

5 A B 3 3A B Solve two equations with two unknowns.

5 A B 3 3A B

3 3A B

8 4B

2 B 5 2A

3 A

3 2

3 1dx

x x

3ln 3 2ln 1x x C

This technique is calledPartial Fractions

1

2

5 3

2 3

xdx

x x

The short-cut for this type of problem is

called the Heaviside Method, after English engineer Oliver Heaviside.

5 3

3 1

x

x x

3 1

A B

x x

Multiply by the common denominator.

5 3 1 3x A x B x

8 0 4A B

1

Let x = - 1

2 B

12 4 0A B Let x = 3

3 A

2

5 3

2 3

xdx

x x

The short-cut for this type of problem is

called the Heaviside Method, after English engineer Oliver Heaviside.

5 3

3 1

x

x x

3 1

A B

x x

5 3 1 3x A x B x

8 0 4A B

1

2 B

12 4 0A B 3 A

3 2

3 1dx

x x

3ln 3 2ln 1x x C

2

6 7

2

x

x

Repeated roots: we must use two terms for partial fractions.

22 2

A B

x x

6 7 2x A x B

6 7 2x Ax A B

6x Ax 7 2A B

6 A 7 2 6 B

7 12 B

5 B

2

6 5

2 2x x

2

3 2

2

2 4 3

2 3

x x x

x x

If the degree of the numerator is higher than the degree of the denominator, use long division first.

2 3 22 3 2 4 3x x x x x 2x

3 22 4 6x x x 5 3x

2

5 32

2 3

xxx x

5 3

23 1

xx

x x

3 2

23 1

xx x

4

(from example one)

Examples

2

3 2 2

To compute these numbers , and we get

3 ( 1) ( )( 1)

1 ( 1)( 1) ( 1)( 1)

A B C

A x x Bx C x

x x x x x x x

3

3Compute .

1dx

x

3 2

3 2

Observe 1 ( 1)( 1). Hence

3 for some numbers , and .

1 1 1

x x x x

A Bx CA B C

x x x x

Example 1

0 1

0 1.

3 2

A B A

A B C B

A C C

2

3 3

3 ( ) ( )

1 1

A B x A B C x A C

x x

3 2

Hence

3 1 2

x 1 1 1

x

x x x

Examples

3

3Compute .

1dx

x

2 2

1 2 1 3 1ln 1

2 1 2 1

xx dx dx

x x x x

Example 1 (cont’d)

22

1 3 1ln 1 ln 1

2 2 1/ 2 3 / 4x x x dx

x

3 2

By the previous computations we now have

3 1 2

x 1 1 1

xdx dx dx

x x x

21 2 1ln 1 ln 1 3 arctan

2 3

xx x x K

Substitute u=x2+x+1 in the first remaining integral and rewrite the last integral.

This expression is the required substitution to finish the computation.

Examples

3

2 2

2 2

1 1

x xx

x x

3

2

2Compute .

1

x xdx

x

Example 2

We can simplify the function to be integrated by performing polynomial division first. This needs to be done whenever possible. We get:

3

2 2

2 2 1 1

1 1 1 1

x xx x

x x x x

Partial fraction decomposition for the remaining rational expression leads to

Now we can integrate

Tacoma Narrow Bridge – November 1940 The bridge had been called "Galloping Gertie" and attracted tourists who wanted to feel the sensation of crossing the rolling center span. Although there had been concerns about the bridge's stability, officials had been so confident that they considered canceling the insurance policies. A new and safer bridge was built in 1950 and is still in use today. It is said that the new nickname is "Sturdy Gertie".

The story of the Tacoma bridge is widely used in engineering, physics and calculus classes to motivate the study of differential equations.