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\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}
\[
\int xe^{x}dx\]

\end{example}

\subsubsection*{Solution}

The game, and it is a game, is to pick the right parts to be \( u \) and \( dv \).
You have to be able to integrate whatever you choose as \( dv \), and you have
to hope that, when you differentiate \( u \), it becomes simpler. Here, you
couldn't ask for more. Set \( u=x \), and then \( dv=e^{x}dx \). I usually
write down the other two pieces: \( du=dx \) and \( v=e^{x} \). Don't bother
with the \( +C \) here when you integrate \( dv \), it won't really change
anything. 

Then, applying the equation above --- which is, on one level, all there is to
integration by parts --- we see:\begin{eqnarray*}
\int x\, e^{x}dx & = & \int u\, dv\\
 & = & u\, v-\int v\, du\\
 & = & x\, e^{x}-\int e^{x}dx\\
 & = & x\, e^{x}-e^{x}+C.
\end{eqnarray*}

\end{document}
