MAT136H1 Lecture Notes - Partial Fraction Decomposition, Antiderivative

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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:




  
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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:

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