# At a Glance - Integration by Partial Fractions

Integration by partial fractions is a technique we can use to integrate rational functions when the degree of the numerator is less than the degree of the denominator. Here's the big picture:

• We start out with an integral whose integrand is a rational function, like

The degree of the numerator must be less than the degree of the denominator.

• We do some sneaky stuff to rewrite the original rational function as a sum of partial fractions:

• We integrate the partial fractions, whose antiderivatives all involve the natural log:

Be Careful: When x occurs in a denominator with a coefficient other than 1, you have to use integration by substitution.

#### Example 1

 Decompose into a sum of the form.

#### Example 2

 Decompose into partial fractions.

#### Example 3

 Decompose  into partial fractions.

 Find

#### Example 5

 Findgiven that

#### Exercise 1

Without a calculator, find

Find

Find the sum.

#### Exercise 4

Decompose into partial fractions.

#### Exercise 5

Decompose into partial fractions.

#### Exercise 6

Decompose into partial fractions.

#### Exercise 7

Decompose into partial fractions.

#### Exercise 8

Decompose into partial fractions.

#### Exercise 9

Decompose into partial fractions.

#### Exercise 10

Decompose into partial fractions.

#### Exercise 11

Decompose into partial fractions.

#### Exercise 12

Decompose into partial fractions.

#### Exercise 13

Decompose into partial fractions.

Integrate.

Integrate.

Integrate.

Integrate.

Integrate.