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
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
Evaluate the definite integral using partial fractions.
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. Then
the numerators are written as:
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: