# 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. |

#### Example 4

Find |

#### Example 5

Find given that |

#### Exercise 1

Without a calculator, find

#### Exercise 2

Find

#### Exercise 3

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.

#### Exercise 14

Integrate.

#### Exercise 15

Integrate.

#### Exercise 16

Integrate.

#### Exercise 17

Integrate.

#### Exercise 18

Integrate.