3.4 Partial fractions

We have seen some techniques that allow us to integrate specific rational functions. For example, we know that

However, we do not yet have a technique that allows us to tackle arbitrary quotients of this type. Thus, it is not immediately obvious how to go about evaluating ${\displaystyle \int \frac{3x}{x^2-x-2}dx.}$ However, we know from material previously developed that

In fact, by getting a common denominator, we see that

Consequently,

In this section, we examine the method of partial fraction decomposition    , which allows us to decompose rational functions into sums of simpler, more easily integrated rational functions. Using this method, we can rewrite an expression such as: $\frac{3x}{x^2-x-2}$ as an expression such as $\frac{1}{x+1}+\frac{2}{x-2}.$

The key to the method of partial fraction decomposition is being able to anticipate the form that the decomposition of a rational function will take. As we shall see, this form is both predictable and highly dependent on the factorization of the denominator of the rational function. It is also extremely important to keep in mind that partial fraction decomposition can be applied to a rational function $\frac{P(x)}{Q(x)}$ only if $\text{deg}\left(P\left(x\right)\right)<\text{deg}\left(Q\left(x\right)\right).$ In the case when $\text{deg}\left(P\left(x\right)\right)\ge \text{deg}\left(Q\left(x\right)\right),$ we must first perform long division to rewrite the quotient $\frac{P(x)}{Q(x)}$ in the form $A(x)+\frac{R(x)}{Q(x)},$ where $\text{deg}\left(R\left(x\right)\right)<\text{deg}\left(Q\left(x\right)\right).$ We then do a partial fraction decomposition on $\frac{R(x)}{Q(x)}.$ The following example, although not requiring partial fraction decomposition, illustrates our approach to integrals of rational functions of the form ${\displaystyle \int \frac{P(x)}{Q(x)}dx,}$ where $\text{deg}\left(P\left(x\right)\right)\ge \text{deg}\left(Q\left(x\right)\right).$

To integrate ${\displaystyle \int \frac{P(x)}{Q(x)}dx,}$ where $\text{deg}\left(P\left(x\right)\right)<\text{deg}\left(Q\left(x\right)\right),$ we must begin by factoring $Q(x).$

Nonrepeated linear factors

If $Q(x)$ can be factored as $\left(a_{1}x+b_1\right)\left(a_{2}x+b_2\right)\text{…}\left(a_{n}x+b_n\right),$ where each linear factor is distinct, then it is possible to find constants $A_1,A_2\text{,…}{A}_n$ satisfying

The proof that such constants exist is beyond the scope of this course.

In this next example, we see how to use partial fractions to integrate a rational function of this type.