From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!

# 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.