%% 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
 \theoremstyle{definition}
  \newtheorem{example}[thm]{Example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
\setcounter{section}{5}
\setcounter{subsection}{3}

\makeatother

\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


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

\maketitle

\subsection{Techniques of integration}

\begin{example}
\[
\int \frac{x^{2}}{x^{3}+7}dx\]
 
\end{example}

\subsubsection*{Solution}

Here, the denominator is the {}``inside{}''. The first example as a bit clearer
on what was the inside term, the prime candidate for \( u \), but as you'll
see denominators are also likely suspects. So, set \[
u=x^{3}+7,\]
and then \begin{eqnarray*}
du & = & 3x^{2}dx,\, \textrm{or}\\
\frac{1}{3x^{2}}du & = & dx.
\end{eqnarray*}
Supstitute in terms of \( u \) and \( du \):

\begin{eqnarray*}
\int \frac{x^{2}}{x^{3}+7}dx & = & \int \frac{x^{2}}{u}\frac{1}{3x^{2}}du\\
 & = & \frac{1}{3}\int \frac{du}{u}\\
 & = & \frac{1}{3}\ln |u|+C.
\end{eqnarray*}
At this stage I always ask myself: {}``Now, what was \( u \)?{}'' \( u=x^{3}+7 \).
so \begin{eqnarray*}
\int \frac{x^{2}}{x^{3}+7}dx & = & \frac{1}{3}\ln |u|+C\\
 & = & \frac{1}{3}\ln \left| x^{3}+7\right| +C.
\end{eqnarray*}

\end{document}
