Integration by Using Partial Fractions

In this tutorial we shall discuss using partial fractions to find the integration of rational functions. We shall illustrate this method with the help of suitable examples in later tutorials.

Rational Function: A function which can be expressed as the quotient of two polynomial functions is called a rational function. Thus, the function $$R$$ is a rational function if
\[R\left( x \right) = \frac{{P\left( x \right)}}{{Q\left( x \right)}}\]

where $$P\left( x \right)$$ and $$Q\left( x \right) \ne 0$$ are polynomials.

If the degree of the numerator is not less than the degree of the denominator we have an improper fraction, and in this case we divide the numerator by denominator until we obtain a proper fraction (one in which the degree of the numerator is less than the degree of the denominator). For example,
\[\frac{{{x^4} – 10{x^2} + 3x + 1}}{{{x^2} – 4}} = {x^2} – 6 + \frac{{3x – 23}}{{{x^2} – 4}}\]

So, if we wish to integrate$$\int {\frac{{{x^4} – 10{x^2} + 3x + 1}}{{{x^2} – 4}}dx} $$ the problem is reduced to integrating $$\int {\frac{{3x – 23}}{{{x^2} – 4}}dx} $$

Generally we shall be concerned with the integration of the expression of the form
\[\int {\frac{{P\left( x \right)}}{{Q\left( x \right)}}dx} \]

where the degree of $$P\left( x \right)$$ is less than the degree of $$Q\left( x \right)$$.

In order to do this it is often necessary to write $$\frac{{P\left( x \right)}}{{Q\left( x \right)}}$$ as the sum of partial fractions. The denominators of the partial fractions are obtained by factoring $$Q\left( x \right)$$ into a product of linear factors and quadratic equations.