In this section, we will look at integrating more complicated rational functions using the technique of partial fraction decomposition.

Section 8.2 Integration By Partial Fraction Decomposition

Idea

The integral seems difficult to evaluate.

The integral is not.

Idea

The integral seems difficult to evaluate.

The integral is not.

They are the same integral!

Idea

The integral seems difficult to evaluate.

The integral is not.

They are the same integral!

How do we convert the first integral into the second?

Terminology

A rational function is called proper if the denominator’s degree > numerator’s degree. Otherwise it is called improper.

Any improper rational function can be rewritten as the sum of a polynomial and a proper rational function by performing polynomial long division.

Any rational function can also be written as a sum of partial fractions – other rational functions having irreducible factors of the original denominator.

For Example

Consider the function .

By going through the long division process, we can rewrite this as:

Fact

All polynomials can be written as a product of linear and irreducible quadratic factors raised to powers.

Thus, all partial fractions will have one of two forms:

The Process

1. Make the integrand proper

2. Factor the denominator completely

3. Write as a sum of partial fractions with undetermined numerator coefficients

4. Algebraically find the value of these coefficients.

5. Antidifferentiate the result fraction by fraction

Example 1

Find

Example 2

Find

Example 3

Find

Example 4

Find

Example 5

Find

Example 6

Find