# MAT136H1 Lecture Notes - Partial Fraction Decomposition, Antiderivative

7.4 Integration Techniques

Partial Fractions for Rational Functions

Question #2 (Medium): Evaluating the Definite Integral Using Partial Fractions

Strategy

Before evaluating the definite integral, rational function needs to be decomposed by partial fractions.

Once the numerator coefficients are determined, taking the anti-derivative is easy. To decompose as

much as possible, factored binomials with power greater than 1 must be written by that number of

times.

Sample Question

Evaluate the definite integral using partial fractions.

Solution

Since the denominator is already factored, coefficients need to be determined to evaluate the integral.

So,

. Since is of order 2, it needs to be written twice. Then

the numerators are written as:

. This

means, , and . Then . Substitute into the

other equations: , so . Likewise,

. Then, . Through elimination,

. So , then . And since ,

, so . Then through partial fraction, the integral is decomposed into:

Therefore using partial fractions, the integral is evaluated as:

## Document Summary

Question #2 (medium): evaluating the definite integral using partial fractions. Before evaluating the definite integral, rational function needs to be decomposed by partial fractions. Once the numerator coefficients are determined, taking the anti-derivative is easy. To decompose as much as possible, factored binomials with power greater than 1 must be written by that number of times. Since the denominator is already factored, coefficients need to be determined to evaluate the integral. Since ( ) is of order 2, it needs to be written twice. So, the numerators are written as: ( )( ) ( ) ( ) ( ) Substitute into the other equations: ( ) , so . Then through partial fraction, the integral is decomposed into: