- Topics At a Glance
- Indefinite Integrals Introduction
- Integration by Substitution: Indefinite Integrals
- Legrange (Prime) Notation
- Leibniz (Fraction) Notation
- Integration by Substitution: Definite Integrals
- Integration by Parts: Indefinite Integrals
- Some Tricks
- Integration by Parts: Definite Integrals
**Integration by Partial Fractions**- Integrating Definite Integrals
- Choosing an Integration Method
- Integration by Substitution
- Integration by Parts
- Integration by Partial Fractions
- Thinking Backwards
- Improper Integrals
- Badly Behaved Limits
- Badly Behaved Functions
- Badly Behaved Everything
- Comparing Improper Integrals
- The
*p*-Test - Finite and Infinite Areas
- Comparison with Formulas
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

**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. *A* and *B* are unknown numbers.

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.