Integration by Partial Fraction Decomposition (with Distinct Linear Factors)

Introduction

The method of partial fractions is a powerful technique used to integrate rational functions in proper form (that is, the degree of the numerator is less than the degree of the denominator), which are expressions of the form \dfrac{f(x)}{g(x)} where f(x) \text{ and } g(x) are polynomials.

Once in proper form, the rational expression is then decomposed into simpler fractions, which can be integrated using basic rules to be explained by the illustrative examples below.

Illustrative Example #1

Evaluate the integral

irst, factor the denominator:

Now decompose the rational function into partial fractions:

Multiply both sides by \(x(x - 1)\) to eliminate the denominators:

Now, solve for \(A\) and \(B\) using convenient values:

Let x=0:

Let x = 1:

So we now have:

Now substitute into the integral:

Integrate each term:

Thus,

Illustrative Example #2

Evaluate the integral \displaystyle \int \frac{5x - 7}{x^2 - 3x + 2} \, dx

First, factor the denominator:

Now decompose the rational function into partial fractions:

Multiply both sides by (x - 1)(x - 2) to eliminate denominators:

Choose convenient values of x to solve for A and B. For the sake of simplicity, we will set x - 1 \text{ and } x - 2 \text{ to } 0:

Let x=2:

Let x=1:

So we have:

Now substitute into the integral:

Since these are basic logarithmic integrals (solvable using $u$-substitution), integrate each term:

Thus,

Illustrative Example #3

Evaluate the integral \displaystyle \int \frac{3x + 13}{2x^2 + 9x + 10} , dx.

First, factor the denominator:

Now decompose the rational function into partial fractions:

Multiply both sides by \((2x + 5)(x + 2)\) to eliminate denominators:

Choose convenient values of x \text{ to solve for } A \text{ and } B. Set the denominators to zero:

Let x=-2:

Let x=-\dfrac{5}{2}:

So we have the partial fraction decomposition:

Now substitute into the integral:

Integrate each term (note the \(2x + 5\) needs substitution):

Thus,