% LaTeX file -- PM 352, Winter 2010, Assignment 5

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\begin{document}
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%%%%%%%   TITLE   %%%%%%
{\noindent\textbf{\large PM 352 \hfill Assignment 5 \hfill Due Monday March 22.}} 

\vskip2em 

Evaluate the following integrals using the residue theorem.  Be careful to specify the
curves that you integrate over, and explain why extraneous terms go to 0.\\

\begin{Ex}

\item $\displaystyle\int_0^{\pi/2} \frac{dx}{a + \sin^2 x} \qfor a>0$.
\hspace{28mm}
2.\ \ $\displaystyle\int_0^\infty \frac{x^2}{(x^2+a^2)^3} \, dx \qfor a > 0$.

\stepcounter{exerci}
\item $\displaystyle\int_0^\infty \frac{\cos x}{a^2 + x^2} \, dx \qfor a > 0$.
\hspace{28mm}
4.\ \ $\displaystyle\int_0^\infty \frac{x^{1/3}}{1+x^2} \, dx$

\stepcounter{exerci}
\item $\displaystyle\int_0^\infty \frac{\log x}{(1+x^2)^2} \, dx$
\hspace{49mm}
6.\ \ $\displaystyle\int_0^\infty \frac{dx}{1 + x^n} \qfor n>1$.\ \   Use Fig.2.

\bigbreak\bigbreak
\noindent\hspace*{-17pt}\textbf{Bonus Problems. \ Please hand in separately.}

\item[A.] $\displaystyle\int_0^\infty \frac{\cos x}{\cosh x} \, dx$ \ \ \ Use Fig.3.
\hspace{31mm}
B.\ \ Compute $\dsum_{n=1}^\infty \dfrac1{n^4}$.

\bigbreak\bigbreak
\noindent\hspace*{-17pt}\textbf{Extra Problems. \ Don't hand in.}
\item[C.] $\displaystyle\int_0^\infty \frac{\log x}{x^2-1} \, dx$ \ \ \ Use Fig.1.
\hspace{32mm}
D.\ \ $\displaystyle\int_0^\infty \frac{\sin^2 x}{x^2} \, dx$. 

\item[E.]  $\displaystyle  \int_0^\infty \frac{\cos x}{a^2 - x^2} \, dx \qfor a= (2n+1)\pi/2$, $n\ge1$. \ \ Use Fig.4.

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\centering
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\end{Ex}
\end{document}