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Worksheet Some Exercises for this section
I included some practice problems that cover some main concepts in this section. You don’t need to turn it in, but I highly encourage you to work on this with your classmates. I may take problems here to be your in-class practice problems, homework problems, and/or exam problems. Reach out to Richard for help if you get stuck or have any questions.
I will only include the final answer to some of the problems for you to check your result. If you want to check your work, talk to Richard and he is happy to discuss the process with you.
Exercise Group. Set up the form of the partial fraction decomposition for the following rational expressions.
1.
\(\dfrac{x - 9}{x^2 - 3x - 18}\)
Solution .
\begin{equation*}
\frac{x - 9}{x^2 - 3x - 18} = \frac{A}{x - 6} + \frac{B}{x + 3}
\end{equation*}
2.
\(\dfrac{x^2 - 3x}{x^3 - 3x^2 - 4x}\)
Solution .
\begin{equation*}
\dfrac{x^2 - 3x}{x^3 - 3x^2 - 4x} = \frac{A}{x} + \frac{B}{x - 4} + \frac{C}{x + 1}
\end{equation*}
3.
\(\dfrac{1}{(x + 1)^2\left(x^2 + 2\right)}\)
Solution .
\begin{equation*}
\dfrac{1}{(x + 1)^2\left(x^2 + 2\right)} = \frac{A}{x + 1} + \frac{B}{(x + 1)^2} + \frac{Cx + D}{x^2 + 2}
\end{equation*}
Exercise Group. Evaluate the following integrals.
4.
\(\displaystyle \int \frac{2x - 1}{x^2 - 5x + 6}\, dx\)
Solution .
\begin{equation*}
5\ln|x - 3| - 3\ln|x - 2| + C
\end{equation*}
5.
\(\displaystyle \int \frac{dx}{(x - 2)(x - 3)(x + 2)}\, dx\)
Solution .
\begin{equation*}
\frac{1}{20}\ln|x + 2| - \frac{1}{4}\ln|x - 2| + \frac{1}{5}\ln|x - 3| + C
\end{equation*}
6.
\(\displaystyle \int_{-1}^1 \frac{x}{(x + 3)^2}\, dx\)
Solution .
\begin{equation*}
\ln(2) - \frac{3}{4} \approx -0.0568528194401
\end{equation*}
Exercise Group. Evaluate the following integrals by first making a substitution to convert the integrand into a rational expression.
7.
\(\displaystyle \int \frac{e^x\, dx}{\left(e^x - 1\right)\left(e^x + 2\right)}\)
Solution .
\begin{equation*}
\frac{1}{3}\ln\left|\frac{e^x - 1}{e^x + 2}\right| + C
\end{equation*}
8.
\(\displaystyle \int \frac{e^x\, dx}{e^{2x} - e^x}\)
Solution .
\begin{equation*}
\ln\left|e^x - 1\right| - x + C
\end{equation*}
9.
\(\displaystyle \int \frac{\sqrt{x}\, dx}{x - 1}\)
Solution .
\begin{equation*}
2\sqrt{x} - \ln\left(\sqrt{x} + 1\right) + \ln\left|\sqrt{x} - 1\right| + C
\end{equation*}
