Integration by Partial Fractions

Integration by Partial Fraction Decomposition is a procedure where we can “decompose” a rational function into simpler rational functions that are more easily integrated. So basically, we are breaking up one “complicated” fraction into several different “less complicated” fractions. You may have learned how to use this technique in your Algebra class, and it’s quite useful in Calculus!

The basic idea is to factor the denominator (if it isn’t already factored) of the complicated factor, and then break it up into different fractions with denominators of those factors.

For example, let’s integrate \(\int{{\frac{1}{{12{{x}^{2}}+x-1}}dx}}\):

Since it looks pretty impossible to integrate the way it, let’s see if we can divide it into two separate fractions.  Note that \(\int{{\frac{1}{{12{{x}^{2}}+x-1}}dx}}=\int{{\frac{1}{{\left( {3x+1} \right)\left( {4x-1} \right)}}dx}}\). (When doing these types of problems in school, the problem will probably be easy to factor; in the “real world,” this would probably not be the case.)