# MAT136H1 Lecture Notes - Partial Fraction Decomposition

7.4 Integration Techniques

Partial Fractions for Rational Functions

Question #1 (Easy): Decompose Rational Function into Partial Fractions

Strategy

Rational function with factorable denominator can be decomposed into partial fractions. Combining

partial fractions reverses the process and result in the original rational function. Step 1: factor the

denominator. Step 2: write partial fractions with unknown coefficients at numerator. Step 3: solve for

the coefficients.

Sample Question

Write the function into partial fraction decomposition.

Solution

The denominator can be factored: . This means

the given function can be decomposed into 3 partial fractions:

. Combine the

numerators: . Simplify the left side:

Combine the like powers: .

Then and and

Solve simultaneously, , and

It is like solving a system of linear equations.

Use substitution , then , then

, so . Then since ,

Thus,

By decomposition, the rational function is equal to the partial fraction:

## Document Summary

Question #1 (easy): decompose rational function into partial fractions. Rational function with factorable denominator can be decomposed into partial fractions. Combining partial fractions reverses the process and result in the original rational function. Step 2: write partial fractions with unknown coefficients at numerator. The denominator can be factored: ( ) ( )( ). This means the given function can be decomposed into 3 partial fractions: Combine the numerators: ( )( ) ( ) ( ) . Combine the like powers: ( ) ( ) . It is like solving a system of linear equations. Use substitution , then ( ) , then.