Rational Functions Integrals

Calculate integrals of rational functions using partial fractions decomposition: Examples and detailed solutions, more questions and their solutions are included. 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.
An online partial fractions decomposition calculator may be used to decompose rational functions.

Examples with Solutions

Example 1

Evaluate the integral
\( \displaystyle \int \dfrac{-5x + 11}{x^2+x-2} dx \)
Solution to Example 1:
We use partial fractions decomposition to decompose the integrand into simpler fractions.
\( \dfrac{-5x + 11}{x^2+x-2} = \dfrac{2}{x-1} - \dfrac{7}{x+2} \)

We now use table of integrals to integrate

\( \displaystyle \int \dfrac{-5x + 11}{x^2+x-2} dx = \int \dfrac{2}{x-1} dx - \int \dfrac{7}{x+2} dx = 2 \ln|x - 1| - 7 \ln|x+2| + C \)

Example 2

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

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
\( \displaystyle \int \dfrac{x^2+6x - 3}{(x+3)(x^2+2x+9)} dx = \int \dfrac{2x+2}{x^2+2x+9} dx - \int \dfrac{1}{x+3} dx \\ = \ln|x^2+2x+9| - \ln|x+3| + C \)


Example 3

Evaluate the integral
\( \displaystyle \int \dfrac{2x^3 + 10x^2 +11x}{x^2+5x+6} dx \)

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:
\( \dfrac{2x^3 + 10x^2 +11x}{x^2+5x+6} = 2x - \dfrac{x}{x^2+5x+6} \)

A partial fractions decomposition of the term \( \dfrac{x}{x^2+5x+6} \) 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
\( \displaystyle \int \dfrac{2x^3 + 10x^2 +11x}{x^2+5x+6} dx = \int 2x dx +\int \dfrac{2}{x+2} dx - \int \dfrac{3}{x+3} dx \)

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

Questions

Evaluate the following integrals.
1. \( \displaystyle \int \dfrac{-x + 7}{x^2+x-2} dx \)

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

3. \( \displaystyle \int \dfrac{x^4+3x^3+2x^2+7x+9}{x^2+3x+2} dx \)

Solutions to Above Questions

1.
Decompose into simpler fraction: \( \dfrac{-x + 7}{x^2+x-2} = \dfrac{2}{x-1} - \dfrac{3}{x+2} \)
Hence
\( \displaystyle \int \dfrac{-x + 7}{x^2+x-2} dx = \int (\dfrac{2}{x-1} - \dfrac{3}{x+2}) dx \\ = 2 \ln|x-1| - 3 \ln|x+2| + C \)

2.
Decompose into simpler fractions: \( \dfrac{-8x^2 +23x - 5}{(x+7)(2x^2+x+2)} = -\dfrac{6}{x+7} + \dfrac{4x+1}{2x^2+x+2} \)

Hence \( \displaystyle \int \dfrac{-8x^2 +23x - 5}{(x+7)(2x^2+x+2)} dx = (-\dfrac{6}{x+7} + \dfrac{4x+1}{2x^2+x+2} ) dx \)

\( \qquad = \ln|2x^2+x+2| - 6 \ln|x+7| + C \)

3.
Decompose into simpler fractions: \( \dfrac{x^4+3x^3+2x^2+7x+9}{x^2+3x+2} = x^2 + \dfrac{2}{x+1} + \dfrac{5}{x+2} \)

Hence \( \displaystyle \int \dfrac{x^4+3x^3+2x^2+7x+9}{x^2+3x+2} dx = ( x^2 + \dfrac{2}{x+1} + \dfrac{5}{x+2} ) dx \)

\( \qquad = \dfrac{x^3}{3}+2\ln |x+1|+5 \ln |x+2| + C \)

More references on integrals and their applications in calculus.
online partial fractions decomposition calculator
partial fractions decomposition