Calculate integrals of rational functions using partial fractions decomposition. This tutorial presents examples with detailed solutions, including instances where the degree of the numerator is greater than or equal to the degree of the denominator.
Note: In what follows, \( C \) represents the constant of integration.
You can also use an online partial fractions decomposition calculator to help decompose rational functions.
Examples with Solutions
Click on each example to view its detailed step-by-step solution.
Example 1: Evaluate \( \displaystyle \int \dfrac{-5x + 11}{x^2 + x - 2} \, dx \)
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 evaluate the integral term by term:
\[ \int \dfrac{-5x + 11}{x^2 + x - 2} \, dx = \int \dfrac{2}{x - 1} \, dx - \int \dfrac{7}{x + 2} \, dx \] \[ = 2\ln|x - 1| - 7\ln|x + 2| + C \]Example 2: Evaluate \( \displaystyle \int \dfrac{x^2 + 6x - 3}{(x + 3)(x^2 + 2x + 9)} \, dx \)
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 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 \( \displaystyle \int \dfrac{2x^3 + 10x^2 + 11x}{x^2 + 5x + 6} \, dx \)
Solution to Example 3:
In this example, the degree of the numerator is greater than the degree of the denominator. Polynomial long division is carried out first:
\[ \dfrac{2x^3 + 10x^2 + 11x}{x^2 + 5x + 6} = 2x - \dfrac{x}{x^2 + 5x + 6} \]A partial fractions decomposition of the term \( \dfrac{x}{x^2 + 5x + 6} \) gives:
\[ \dfrac{2x^3 + 10x^2 + 11x}{x^2 + 5x + 6} = 2x + \dfrac{2}{x + 2} - \dfrac{3}{x + 3} \]Using the above, the integral may be written as:
\[ \int \dfrac{2x^3 + 10x^2 + 11x}{x^2 + 5x + 6} \, dx = \int 2x \, dx + \int \dfrac{2}{x + 2} \, dx - \int \dfrac{3}{x + 3} \, dx \]Evaluating the integrals yields:
\[ = x^2 + 2\ln|x + 2| - 3\ln|x + 3| + C \]Practice Exercises
Evaluate the following integrals using partial fractions. Click each exercise to reveal the step-by-step solution.
Exercise 1: Evaluate \( \displaystyle \int \dfrac{-x + 7}{x^2 + x - 2} \, dx \)
Solution:
Decompose into simpler fractions:
\[ \dfrac{-x + 7}{x^2 + x - 2} = \dfrac{2}{x - 1} - \dfrac{3}{x + 2} \]Hence,
\[ \int \dfrac{-x + 7}{x^2 + x - 2} \, dx = \int \left(\dfrac{2}{x - 1} - \dfrac{3}{x + 2}\right) dx \] \[ = 2\ln|x - 1| - 3\ln|x + 2| + C \]Exercise 2: Evaluate \( \displaystyle \int \dfrac{-8x^2 + 23x - 5}{(x + 7)(2x^2 + x + 2)} \, dx \)
Solution:
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,
\[ \int \dfrac{-8x^2 + 23x - 5}{(x + 7)(2x^2 + x + 2)} \, dx = \int \left(-\dfrac{6}{x + 7} + \dfrac{4x + 1}{2x^2 + x + 2}\right) dx \] \[ = \ln|2x^2 + x + 2| - 6\ln|x + 7| + C \]Exercise 3: Evaluate \( \displaystyle \int \dfrac{x^4 + 3x^3 + 2x^2 + 7x + 9}{x^2 + 3x + 2} \, dx \)
Solution:
Because the degree of the numerator is higher, perform polynomial division first, then decompose:
\[ \dfrac{x^4 + 3x^3 + 2x^2 + 7x + 9}{x^2 + 3x + 2} = x^2 + \dfrac{2}{x + 1} + \dfrac{5}{x + 2} \]Hence,
\[ \int \dfrac{x^4 + 3x^3 + 2x^2 + 7x + 9}{x^2 + 3x + 2} \, dx = \int \left(x^2 + \dfrac{2}{x + 1} + \dfrac{5}{x + 2}\right) dx \] \[ = \dfrac{x^3}{3} + 2\ln|x + 1| + 5\ln|x + 2| + C \]