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\title{\vspace*{-1.5in}\textbf{\S 5.10: Improper Integrals}}
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\begin{document}

\maketitle
%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300--007: Calculus II}
\rhead{February 2, 2018}
\chead{Improper Integrals}
\vspace{-.5in}
\noindent
Decide whether or not the following improper integrals converge or diverge. 
\vskip12pt
\noindent \framebox{\textbf{Type I:} Integrals over infinite intervals}
\begin{enumerate}
 \item $\displaystyle \int_1^\infty \frac{1}{x}\,dx$ \vskip6cm
 \item $\displaystyle  \int_3^\infty\frac{1}{9+x^2}\,dx$ \vskip6cm
 \item $\displaystyle \int_1^\infty\frac{1}{x^p}\,dx$, where $p \neq 1$\vskip6cm
\end{enumerate}
\newpage
\noindent \framebox{\textbf{Type II:} Integrals of functions with vertical asymptotes}
\begin{enumerate}
\setcounter{enumi}{3}
 \item $\displaystyle \int_1^2\frac{1}{\sqrt{x-1}}\,dx$\vskip6cm
 \item $\displaystyle \int_0^1\ln(x)\,dx$\vskip6cm
 \item $\displaystyle \int_1^4\frac{1}{x-2}\,dx$\vskip6cm
\end{enumerate}
\newpage
\noindent\framebox{\textbf{Miscellaneous}}
\begin{enumerate}
\setcounter{enumi}{6}
\item $\displaystyle \int_{-\infty}^\infty te^{-t^2}\,dt$\vskip6cm
\item $\displaystyle \int_1^\infty\frac{1}{x\ln(x)}\,dx$\vskip6cm
\item $\displaystyle \int_{-\infty}^{10}\sin^2{x}\,dx  $\vskip6cm
\end{enumerate}
\newpage
\noindent\framebox{\textbf{Comparison Test}}
\begin{enumerate}
\setcounter{enumi}{9}
\item $\displaystyle \int_3^\infty \frac{\ln(x)}{\sqrt{x}}\,dx$\vskip9cm
\item $\displaystyle \int_1^{\infty} \frac{|\sin{x}|}{x^2 + 1}\,dx$\vskip9cm
\end{enumerate}
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