# Active Calculus

## Section9.2Integrating Rational Functions

### Subsection9.2.1Preview Activity

#### Question9.2.1.

For each of the indefinite integrals below, decide whether the integral can be evaluated: immediately because it's a basic integral, using $$u$$-substitution, integration by parts, with multiple techniques, or if none of those will work.
$$\displaystyle \int \frac{1}{1+x^2} \, dx$$
• Basic Integral
• u-Substitution
• Integration by Parts
• A combination of techniques
• None of the above
$$\displaystyle \int \frac{x}{1+x^2} \, dx$$
• Basic Integral
• u-Substitution
• Integration by Parts
• A combination of techniques
• None of the above
$$\displaystyle \int \frac{2x+3}{1+x^2} \, dx$$
• Basic Integral
• u-Substitution
• Integration by Parts
• A combination of techniques
• None of the above
$$\displaystyle \int \frac{e^x}{1+(e^x)^2} \, dx$$
• Basic Integral
• u-Substitution
• Integration by Parts
• A combination of techniques
• None of the above
$$\displaystyle \int \frac{1}{x^2-1} \, dx$$
• Basic Integral
• u-Substitution
• Integration by Parts
• A combination of techniques
• None of the above
Hint.
Note that $$\displaystyle \frac{2x+3}{1+x^2}$$ can be rewritten as $$\displaystyle \frac{2x}{1+x^2} +\frac{3}{1+x^2}\text{.}$$

#### Question9.2.2.

First, note that we can use algebra to combine $$\frac{-5}{x+1}+\frac{6}{x-1}$$ into one fraction by finding a common denominator.
\begin{equation*} \frac{-5}{x+1}+\frac{6}{x-1} = \frac{-5}{x+1}\cdot\frac{x-1}{x-1}+\frac{6}{x-1}\cdot \frac{x+1}{x+1}=\frac{x+11}{x^{2}-1} \end{equation*}
Also note that we know how to integrate
$$\displaystyle \int \frac{-5}{x+1} \, dx =$$ and
$$\displaystyle \int \frac{6}{x-1} \, dx =$$
Which means we can integrate both sides of $$\frac{-5}{x+1}+\frac{6}{x-1} = \frac{x+11}{x^{2}-1}$$ to get
$$\displaystyle \int \frac{x+11}{x^{2}-1} \, dx =$$
Hint.
Remember that you really should use absolute value instead of parentheses when you integrate to get $$\ln\text{.}$$ Sometimes WeBWorK doesn't care about the difference, but sometimes it does. For example, $$\int \frac{1}{x+11} \, dx = \ln |x+11| +C\text{,}$$ not $$\ln(x+11) +C\text{.}$$

#### Question9.2.3.

Find the quotient and remainder using long division for
\begin{equation*} \frac{x^5- x^4+ 3 x^3 - 3 x^2+ 5 x - 8 }{x-1}. \end{equation*}
The quotient is
The remainder is

#### Question9.2.4.

Perform the indicated division and write the quotient and remainder in the provided blanks.
\begin{equation*} \frac{3 x^2 - x - 6}{x-1} \end{equation*}
Answer: $$+$$ $$/ (x - 1)$$

### Exercises9.2.2Exercises

#### 1.

Which of the following is the correct form of the partial fraction decomposition of $$\displaystyle \frac{x-1}{x^3+x^2}?$$
• $$\displaystyle \displaystyle \frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+1}$$
• $$\displaystyle \displaystyle \frac{Ax+B}{x}+\frac{Cx+D}{x^2}+\frac{Ex+F}{x+1}$$
• $$\displaystyle \displaystyle \frac{Ax+B}{x}+\frac{Cx+D}{x^2}+\frac{Ex+F}{x+1}$$
• $$\displaystyle \displaystyle \frac{Ax+B}{x}+\frac{Cx+D}{x^2}+\frac{E}{x+1}$$

#### 2.

Which of the following is the correct form of the partial fraction decomposition of $$\displaystyle \frac{x-1}{x^3+x}?$$
• $$\displaystyle \displaystyle \frac{A}{x}+\frac{Bx+C}{x^2+1}$$
• $$\displaystyle \displaystyle \frac{A}{x}+\frac{B}{x^2+1}$$
• $$\displaystyle \displaystyle \frac{Ax+B}{x}+\frac{Cx+D}{x^2+1}$$
• $$\displaystyle \displaystyle \frac{Ax+B}{x}+\frac{C}{x^2+1}$$

#### 3.

What is the correct form of the partial fraction decomposition for the following integral?
\begin{equation*} \int \frac{x - 11}{x^3 + 11 x^2 - 12 x} \, dx \end{equation*}
• $$\displaystyle \displaystyle \int \left( \frac{A x + B}{x} + \frac{C x + D}{x - 1} + \frac{E x + F}{x + 12} \right)\, dx$$
• $$\displaystyle \displaystyle \int \left( \frac{A}{x} + \frac{B x + C}{x - 11} + \frac{D x + E}{x + 1} \right)\, dx$$
• $$\displaystyle \displaystyle \int \left( \frac{A}{x} + \frac{B}{x - 11} + \frac{C}{x + 1} \right)\, dx$$
• There is no partial fraction decomposition because the denominator does not factor.
• There is no partial fraction decomposition yet because there is cancellation.
• $$\displaystyle \displaystyle \int \left( \frac{A}{x - 11} + \frac{B}{x - 12} + \frac{C}{x + 1} \right)\, dx$$
• There is no partial fraction decomposition yet because long division must be done first.
• $$\displaystyle \displaystyle \int \left( \frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x + 12} \right)\, dx$$

#### 4.

What is the correct form of the partial fraction decomposition for the following integral?
\begin{equation*} \int \frac{13 (x^{6} + 8)}{(x - 9)(x^2 - 1)^2 (x^2 + 9)^2} \, dx \end{equation*}
• $$\displaystyle \displaystyle \int \left( \frac{A}{x - 9} + \frac{Bx+C}{x^2-1} + \frac{Cx+D}{(x^2-1)^2} + \frac{Ex+F}{x^2 + 9} + \frac{Gx+H}{(x^2 + 9)^2} \right)\, dx$$
• There is no partial fraction decomposition yet because long division must be done first.
• $$\displaystyle \displaystyle \int \left( \frac{A}{x - 9} + \frac{B}{(x^2-1)^2} + \frac{C}{(x^2 + 9)^2} \right)\, dx$$
• $$\displaystyle \displaystyle \int \left( \frac{A}{x - 9} + \frac{B}{x+1} + \frac{C}{(x+1)^2} + \frac{D}{x-1} + \frac{E}{(x-1)^2} + \frac{Fx+G}{x^2 + 9} + \frac{Hx+I}{(x^2 + 9)^2} \right)\, dx$$
• There is no partial fraction decomposition yet because there is cancellation.
• $$\displaystyle \displaystyle \int \left( \frac{A}{x - 9} + \frac{B}{x+1} + \frac{C}{(x+1)^2} + \frac{D}{x-1} + \frac{E}{(x-1)^2} + \frac{Fx+G}{x^2 + 9} \right)\, dx$$
• There is no partial fraction decomposition because the denominator does not factor.
• $$\displaystyle \displaystyle \int \left( \frac{A}{x - 9} + \frac{Bx+C}{(x^2-1)^2} + \frac{Dx+E}{(x^2 + 9)^2} \right)\, dx$$

#### 5.

Consider the indefinite integral $$\displaystyle \int \frac{6 x^3+3 x^2 - 144 x - 65}{x^2-25}\, dx$$
Then the integrand decomposes into the form
\begin{equation*} a x + b + \frac{c}{x-5} +\frac{d}{x+5} \end{equation*}
where
$$a$$ =
$$b$$ =
$$c$$ =
$$d$$ =
Integrating term by term, we obtain that
$$\displaystyle \int \frac{6 x^3+3 x^2 - 144 x - 65}{x^2-25}\, dx =$$
$$+C$$

#### 6.

Evaluate the integral
\begin{equation*} \int {\frac{-2}{(x+a)(x+b)}}\, dx \end{equation*}
for the cases where $$a=b$$ and where $$a \ne b.$$
Note: For the case where $$a=b,$$ use only $$a$$ in your answer. Also, use an upper-case "C" for the constant of integration.
If $$a=b:$$
If $$a \ne b:$$

#### 7.

Consider the integral
\begin{equation*} \int \frac{x^{21}-7 x^{14}+5 x^7-39}{\left(x^3-5 x^2+4 x\right)^3 \left(x^4-256\right)^2} \,dx \end{equation*}
Enter a T or an F in each answer space below to indicate whether or not a term of the given type occurs in the general form of the complete partial fractions decomposition of the integrand. $$A_1, A_2, A_3\dots$$ and $$B_1, B_2, B_3,\dots$$ denote constants.
$$\int \frac{2x}{x^{2}-9}\,dx = \int$$ $$dx + \int$$ $$dx =$$ + $$+ C\text{.}$$
(Note that you should not include the $$+C$$ in your entered answer, as it has been provided at the end of the expression.)
Next, use the substitution $$w = x^2 - 9$$ to find the integral:
$$\int \frac{2x}{x^{2}-9}\,dx = \int$$ $$dw =$$ $$+ C =$$ $$+ C\text{.}$$
(For the second answer blank, give your antiderivative in terms of the variable $$w\text{.}$$ Again, note that you should not include the $$+C$$ in your answer.)