## Partial Fractions With Linear Factors

### Learning Outcomes

• Integrate a rational function using the method of partial fractions
• Recognize simple linear factors in a rational function
• Recognize repeated linear factors in a rational function

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

$\displaystyle\int \frac{du}{u}=\text{ln}|u|+C\text{ and }\displaystyle\int \frac{du}{{u}^{2}+{a}^{2}}=\frac{1}{a}{\tan}^{-1}\left(\frac{u}{a}\right)+C\text{.}$

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

$\displaystyle\int \left(\frac{1}{x+1}+\frac{2}{x - 2}\right)dx=\text{ln}|x+1|+2\text{ln}|x - 2|+C$.

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

$\frac{1}{x+1}+\frac{2}{x - 2}=\frac{3x}{{x}^{2}-x - 2}$.

Consequently,

$\displaystyle\int \frac{3x}{{x}^{2}-x - 2}dx=\displaystyle\int \left(\frac{1}{x+1}+\frac{2}{x - 2}\right)dx$.

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 $\frac{P\left(x\right)}{Q\left(x\right)}$ 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 $\frac{P\left(x\right)}{Q\left(x\right)}$ in the form $A\left(x\right)+\frac{R\left(x\right)}{Q\left(x\right)}$, 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 $\frac{R\left(x\right)}{Q\left(x\right)}$. The following example, although not requiring partial fraction decomposition, illustrates our approach to integrals of rational functions of the form $\displaystyle\int \frac{P\left(x\right)}{Q\left(x\right)}dx$, where $\text{deg}\left(P\left(x\right)\right)\ge \text{deg}\left(Q\left(x\right)\right)$.

### Example: Integrating $\displaystyle\int \frac{P\left(x\right)}{Q\left(x\right)}dx$, where $\text{deg}\left(P\left(x\right)\right)\ge \text{deg}\left(Q\left(x\right)\right)$

Evaluate $\displaystyle\int \frac{{x}^{2}+3x+5}{x+1}dx$.

### Recall: Polynomial Long Division

1. Set up the division problem as the numerator divided by the denominator
2. Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.
3. Multiply the answer by the divisor and write it below the like terms of the dividend.
4. Subtract the bottom binomial from the top binomial.
5. Bring down the next term of the dividend.
6. Repeat steps 2–5 until reaching the last term of the dividend.
7. If the remainder is non-zero, express as a fraction using the divisor as the denominator.

### try it

Evaluate $\displaystyle\int \frac{x - 3}{x+2}dx$.

Watch the following video to see the worked solution to the above Try It

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.

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

## Nonrepeated Linear Factors

If $Q\left(x\right)$ can be factored as $\left({a}_{1}x+{b}_{1}\right)\left({a}_{2}x+{b}_{2}\right)\ldots\left({a}_{n}x+{b}_{n}\right)$, where each linear factor is distinct, then it is possible to find constants ${A}_{1},{A}_{2}\text{,}\ldots {A}_{n}$ satisfying

$\frac{P\left(x\right)}{Q\left(x\right)}=\frac{{A}_{1}}{{a}_{1}x+{b}_{1}}+\frac{{A}_{2}}{{a}_{2}x+{b}_{2}}+\cdots +\frac{{A}_{n}}{{a}_{n}x+{b}_{n}}$.

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.

### Example: Partial Fractions with Nonrepeated Linear Factors

Evaluate $\displaystyle\int \frac{3x+2}{{x}^{3}-{x}^{2}-2x}dx$.

In the next example, we integrate a rational function in which the degree of the numerator is not less than the degree of the denominator.

### Example: Dividing before Applying Partial Fractions

Evaluate $\displaystyle\int \frac{{x}^{2}+3x+1}{{x}^{2}-4}dx$.

As we see in the next example, it may be possible to apply the technique of partial fraction decomposition to a nonrational function. The trick is to convert the nonrational function to a rational function through a substitution.

### Example: Applying Partial Fractions after a Substitution

Evaluate $\displaystyle\int \frac{\cos{x}}{{\sin}^{2}x-\sin{x}}dx$.

### try it

Evaluate $\displaystyle\int \frac{x+1}{\left(x+3\right)\left(x - 2\right)}dx$.

Watch the following video to see the worked solution to the above Try It

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.

## Repeated Linear Factors

For some applications, we need to integrate rational expressions that have denominators with repeated linear factors—that is, rational functions with at least one factor of the form ${\left(ax+b\right)}^{n}$, where $n$ is a positive integer greater than or equal to $2$. If the denominator contains the repeated linear factor ${\left(ax+b\right)}^{n}$, then the decomposition must contain

$\frac{{A}_{1}}{ax+b}+\frac{{A}_{2}}{{\left(ax+b\right)}^{2}}+\cdots +\frac{{A}_{n}}{{\left(ax+b\right)}^{n}}$.

As we see in our next example, the basic technique used for solving for the coefficients is the same, but it requires more algebra to determine the numerators of the partial fractions.

### Example: Partial Fractions with Repeated Linear Factors

Evaluate $\displaystyle\int \frac{x - 2}{{\left(2x - 1\right)}^{2}\left(x - 1\right)}dx$.

### Try It

Set up the partial fraction decomposition for $\displaystyle\int \frac{x+2}{{\left(x+3\right)}^{3}{\left(x - 4\right)}^{2}}dx$. (Do not solve for the coefficients or complete the integration.)

Watch the following video to see the worked solution to the above Try It

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.