Integration Using Partial Fractions

Calculate integrals of rational functions using partial fractions decomposition: Tutorials with examples and detailed solutions, exercises with answers are also presented at the end of the page. Examples with degree of numerator greater greater than or equal to the degree of the denominator are also included. In what follows, C is the constant of integration.

Example 1: Evaluate the integral

\int \dfrac{-5x + 11}{x^2+x-2} dx
Solution to Example 1:
We use partial fractions decomposition to decompose the integrand into simpler fractions whose integrals may be easily found.
\dfrac{-5x + 11}{x^2+x-2} = \dfrac{2}{x-1} - \dfrac{7}{x+2}

We now use table of integrals to integrate
\int \dfrac{-5x + 11}{x^2+x-2} dx = \int \dfrac{2}{x-1} dx - \int \dfrac{7}{x+2} dx = 2 \ln|x - 1| - 7 \ln|x+2| + C


Example 2: Evaluate the integral

\int \dfrac{x^2+6x - 3}{(x+3)(x^2+2x+9)} dx

Solution to Example 2:
A partial fractions decomposition of the integrand gives
\dfrac{x^2+6x - 3}{(x+3)(x^2+2x+9)} = \dfrac{2x+2}{x^2+2x+9} - \dfrac{1}{x+3}

We now use a table of integrals to evaluate the integrals
\int \dfrac{x^2+6x - 3}{(x+3)(x^2+2x+9)} dx = \int \dfrac{2x+2}{x^2+2x+9} dx - \int \dfrac{1}{x+3} dx \\\\\\ = \ln|x^2+2x+9| - \ln|x+3| + C


Example 3: Evaluate the integral

\int \dfrac{2x^3 + 10x^2 +11x}{x^2+5x+6} dx

Solution to Example 3:
In this example the degree of the numerator is greater that the degree of the denominator and therefore a division of the numerator by the denominator is carried out in order to write the integrand as follows:
\dfrac{2x^3 + 10x^2 +11x}{x^2+5x+6} = 2x - \frac{x}{x^2+5x+6}

A partial fractions decomposition of the term x / (x2 + 5 x + 6) gives
\dfrac{2x^3 + 10x^2 +11x}{x^2+5x+6} = 2x - \frac{x}{x^2+5x+6} = 2x +\dfrac{2}{x+2}-\dfrac{3}{x+3}

Using the above, the given integral may be written as
\int \dfrac{2x^3 + 10x^2 +11x}{x^2+5x+6} dx = \int 2x dx +\int \dfrac{2}{x+2} dx - \int \dfrac{3}{x+3} dx

Using a table of integrals, we evaluate the integrals as follows:
= x^2 + 2 \ln|x+2| - 3 \ln|x+3| + C

Exercises: Evaluate the following integrals.
1.

\int \dfrac{-x + 7}{x^2+x-2} dx

2.
\int \dfrac{-8x^2 +23x - 5}{(x+7)(2x^2+x+2)} dx

3.
\int \frac{x^4+3x^3+2x^2+7x+9}{x^2+3x+2} dx

Answers to Above Exercises
1.
2 \ln|x-1| - 3 \ln|x+2| + C

2.
\ln|2x^2+x+2| - 6 \ln|x+7| + C
3.
\frac{x^3}{3}+2\ln |x+1|+5 \ln |x+2| + C

More references on
integrals and their applications in calculus.