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\begin{document}
\begin{center}

\textbf{College of the Holy Cross, Spring 2018} \\
\textbf{Math 134 Midterm Exam 3} \\
\textbf{Friday, April 13}
\end{center}

\vskip 100pt

\noindent\textbf{Your Name}: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\vskip 200pt


\noindent\textbf{Instructions}:  For full credit, you must show 
{\it all work} on the test pages and place your final answer
in the box provided for the problem.  Use the back of the
preceding page if you need more space for scratch work.  The numbers
next to each part of the questions are their point values.

\vskip 30pt


\noindent\textbf{Please do not write in the space below}

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss \\
\hline\hline
I  & \hfil/ 30\  \hfil \\
\hline
II  & \hfil/ 20\  \hfil \\
\hline
III  & \hfil/ 40\  \hfil \\
\hline\hline
IV  & \hfil/ 10\  \hfil \\
\hline\hline
Total   & \hfil/100\hfil \\
\hline
\end{tabular}
\end{center}

\vfill\eject
\noindent
I.  Compute each of the following integrals using the trigonometric reduction formulas (on separate sheet)
and other methods as appropriate.
\begin{enumerate}
\item[A.] (15) $\displaystyle\int \sin^4 (5x)\ dx$
\vskip 3.0truein
\MyAnswer{Integral $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[B.] (15) $\displaystyle\int x \sec^4 (3x^2) \ dx$
\vskip 3.0truein
\MyAnswer{Integral $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\vfill\eject
\noindent
II.  (Trigonometric Substitutions)
\begin{enumerate}
\item[A.]  (10) What trigonometric substitution would you use to evaluate
$$\int \frac{x^2}{\sqrt{64 + x^2}}\ dx?$$
Make the substitution and simplify so there is no square root in the resulting trigonometric
integral.  
\vskip 3.0truein
\MyAnswer{Integral $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[B.]  (10) Suppose you had used the trig substitution $x = 3\sin\theta$ and you integrated
the resulting trigonometric integral to get
$$\ln |\csc\theta - \cot\theta| + C$$
What is the integral, expressed as a function of $x$?  
\vskip 2.0truein
\MyAnswer{Integral $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\vfill\eject
\noindent
III.  (Partial Fractions)
\begin{enumerate}
\item[A.] (5) To integrate $\displaystyle\frac{x^4 + 1}{x^3 + 7x + 1}$, what would 
be the first step?  Carry the step out and give the rational function to which you would apply the 
partial fraction decomposition.
\vglue 4.0truein
\item[B.] (10) To decompose $\displaystyle\frac{1}{(x+4)^2(x + 1)^3(x^2 + 9)}$ into partial fractions, what would the form of the fractions be (leave coefficients as undetermined; 
\emph{do not try to solve for them}).  
\end{enumerate}
\vfill\eject

\begin{enumerate}
\item[C.]  (10)  Determine the values $A,B,C$ making 
$$\frac{3x + 1}{x(x^2 + 4)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 4}$$
\vglue 3.0truein
\MyAnswer{$A =$}{\qquad\qquad\qquad\qquad} \MyAnswer{$B =$}{\qquad\qquad\qquad\qquad}
\MyAnswer{$C =$}{\qquad\qquad\qquad\qquad} %
\item[D.]  (15)  You have decomposed a rational function $f(x)$ into partial fractions as
$$f(x) = x^2 + 3x + \frac{1}{(x+3)^2} + \frac{2}{(x+3)^3}+ \frac{4x + 3}{x^2 + 1}.$$
What is $\int f(x)\ dx$?
\end{enumerate}
\vskip 150pt
\MyAnswer{Integral $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\vfill\eject
\noindent
IV.  (Improper Integrals)  
\begin{enumerate}
\item[A.] (5)  Is there a finite area between the $x$-axis and the graph 
$y = \frac{1}{x^{2/3}}$ for $0 < x \le 1$?  Set up the appropriate (improper)
integral and determine if it converges.
\vglue 4.0truein

\item[B.] (5)  Does the solid of revolution obtained by rotating the region from part A
about the $x$-axis have a finite volume or not?  Why?
\end{enumerate}
\end{document}

\vskip 200pt 
\MyAnswer{Integral $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\end{document}

