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How to Use Partial Fraction Decomposition for Integration

Partial fraction decomposition has a long name, but it is an easy and simple method. You use the factors in the denominator to create new and nicer-looking fractions. When the new fractions are set, the integration becomes much easier. Nice!

Rule

Instructions for Partial Fraction Decomposition

1.

Factorize the denominator by finding the zeros.

2.

Intermediate calculation (see the box below):

a): Set the expression equal to a sum of fractions and choose the constants $A, B, C, \dots$ in the numerator—one letter for each fraction.
b): Multiply by the common denominator.
c): Sort the different terms individually.
d): Create a system of equations and solve for $A, B, C, \dots$

3.

Insert the result into the integral and integrate. Yay!

Example 1

Compute $\int \frac{3 x}{x^{2} - x - 2} d x$

\begin{array}{l} \int \frac{3 x}{x^{2} - x - 2} d x & = \int \frac{3 x}{(x - 2) (x + 1)} d x \\ \overset{*}{=} \int \frac{2}{x - 2} + \frac{1}{x + 1} d x \\ = 2 \ln | x - 2 | + \ln | x + 1 | + C \end{array}

\begin{array}{l} \int \frac{3 x}{x^{2} - x - 2} d x & = \int \frac{3 x}{(x - 2) (x + 1)} d x \\ \overset{*}{=} \int \frac{2}{x - 2} + \frac{1}{x + 1} d x \\ = 2 \ln | x - 2 | + \ln | x + 1 | + C \end{array}

3x (x − 2)(x + 1) = A x − 2 + B x + 1 3x = A(x + 1) + B(x − 2) 3x = Ax + A + Bx − 2B 3x + 0 = (A + B)x + (A − 2B) Equate terms of the same degree and solve. You then get A + B = 3 and A − 2B = 0, which gives A = 2 and B = 1.

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