%% LyX 1.1 created this file.  For more info, see http://www.lyx.org/.
%% Do not edit unless you really know what you are doing.
\documentclass[11pt,oneside]{amsart}
\usepackage[T1]{fontenc}
\usepackage{geometry}
\geometry{verbose,letterpaper,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
\setlength\parskip{\medskipamount}
\setlength\parindent{0pt}

\makeatletter


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
\providecommand{\LyX}{L\kern-.1667em\lower.25em\hbox{Y}\kern-.125emX\@}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands.
 \theoremstyle{plain}    
 \newtheorem{thm}{Theorem}[section]
 \numberwithin{equation}{section} %% Comment out for sequentially-numbered
 \numberwithin{figure}{section} %% Comment out for sequentially-numbered

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
\setcounter{section}{6}
\setcounter{subsection}{0}

\makeatother

\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{Area between curves}
\vspace{0.3cm}

\[
Area=2\int _{0}^{1}\left( \arctan (x)-\frac{\pi x}{4}\right) dx.\]
Ah, but how do you do that integral? \[
\int \arctan (x)dx\]
 is an integration by parts problem. Take \( u=\arctan (x) \), and \( dv=dx \).
Then \( v=x \) and \( du=dx/(1+x^{2}). \) Then, \begin{eqnarray*}
\int \arctan (x)dx & = & x\cdot \arctan (x)-\int x\frac{dx}{1+x^{2}}\\
 & = & x\cdot \arctan (x)-\int \frac{xdx}{1+x^{2}}.
\end{eqnarray*}
Now do a substitution. Let's not use the letter \( u \) again, though. Substitute
\( w=1+x^{2} \). Then, \( dw=2xdx \), and it just so happens that we have
\( xdx \) to deal with, \( xdx=dw/2 \), so \begin{eqnarray*}
\int \arctan (x)dx & = & x\cdot \arctan (x)-\int \frac{xdx}{1+x^{2}}\\
 & = & x\cdot \arctan (x)-\int \frac{dw/2}{w}\\
 & = & x\cdot \arctan (x)-\frac{1}{2}\ln |w|+C\\
 & = & x\cdot \arctan (x)-\frac{1}{2}\ln \left| 1+x^{2}\right| +C.
\end{eqnarray*}
OK, the absolute value signs are not important. 
\end{document}
