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\begin{document}
	
	\mbox{}
	\title{Problem Set 4: Integral of Rational Functions by Partial Fractions}
	
	\maketitle	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%\section{Introduction} \label{sec:intro}
	\noindent {\footnotesize Note: Most of the problems were taken from the textbook} \cite{jS15}.
	
	\begin{problem}
		Express the following rational functions in partial fraction decomposition without finding the numerical values of the coefficients.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			
			\item $\frac{1}{x^2 + x^4}$
			
			\item $\frac{x^2 - 1}{x^3 + x^2 + x}$
			
			\item $\frac{x^5 + 1}{(x^2 - x)(x^4 + 2x^2 + 1)}$
			
		\end{enumerate}
	\end{problem}
	
	\vspace{3pt}
		
	\begin{problem}
		Evaluate the following integrals:
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			
			\item $\bigintsss \frac{3x - 2}{x+1} \, dx$
			
			\item $\bigintsss \frac{ax}{x^2 - bx} \, dx$
			
			\item $\bigintsss \frac{4x}{x^3 + x^2 + x + 1} \, dx$
			
			\item $\bigintsss \frac{x^3 - 3x + 7}{(x^2 - 4x + 6)^2} \, dx$
		\end{enumerate}
	\end{problem}
	
	\vspace{3pt}
	
	\begin{problem}
		Solve the following integrals.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			
			\item $\bigintsss \frac{\sqrt{1 + \sqrt{x}}}{x} \, dx$
			
			\item $\bigintsss \frac{\sec^2 x}{\tan^2 x + 3 \tan x + 2} \, dx$
			
			\item $\bigintsss \ln(x^2 - x + 2) \, dx$
			
			\item $\bigintsss x \tan^{-1}x \, dx$
		\end{enumerate}
		
	\end{problem}
	
	\vspace{10pt}
	
	\begin{thebibliography}{20}
		
		\bibitem{jS15} J.~Stewart: \emph{Single Variable Calculus} 8th Edition, Cengage Learning, Boston 2015.
		
	\end{thebibliography}
	
\end{document}