MAT136H1 Lecture Notes - Partial Fraction Decomposition

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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:




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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.

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