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\begin{document}
	
	\mbox{}
	\title{Problem Set 1: Integration by Parts}
	
	\maketitle
	
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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	%\section{Introduction} \label{sec:intro}
	\noindent {\footnotesize Note: Most of the problems were taken from the textbook} \cite{jS15}.
		
	\begin{problem}
		Evaluate the following integrals:
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $\bigintsss (\ln 4x)^2 \, dx$
			
			\item $\bigintsss x \sinh x \, dx$
			
			\item $\bigintsss_0^1 \frac{x}{e^{2x}} \, dx$
			
			\item $\bigintsss x7^x \, dx$
			
			\item $\bigintsss_1^{\sqrt{3}} \tan^{-1}(1/x) \, dx$
			
			\item $\bigintsss x^3 e^{x^2} \, dx$
			
			\item $\bigintsss \sin \sqrt{x} \, dx$
			
			\item $\bigintsss_1^2 x^4(\ln x)^2 \, dx$
		\end{enumerate}
	\end{problem}
	
	\begin{problem}
		Use substitution and then integration by parts to evaluate the following integrals:
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $\bigintsss e^{\sqrt{x}} \, dx$
			\item $\bigintsss \frac{\arcsin(\ln x)}{x} \, dx$
		\end{enumerate}
	\end{problem}
	
	\begin{problem}
		For each natural $n$, show that
		\[
			\int_0^{\pi/2} \sin^{2n}x \, dx = \frac{1 \cdot 3 \cdot 5\cdots (2n-1)}{1 \cdot 2 \cdot 4 \cdots 2n} \frac{\pi}{2}.
		\]
	\end{problem}
	
	\vspace{20pt}
	
	\begin{thebibliography}{20}
		
		\bibitem{jS15} J.~Stewart: \emph{Single Variable Calculus} 8th Edition, Cengage Learning, Boston 2015.
		
	\end{thebibliography}
	
\end{document}