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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}
\[
\int x^{2}\sin (x)dx.\]
 
\end{example}

\subsubsection*{Solution}

This one needs two applications of the trick. First, set \( u=x^{2} \) and
\( dv=\sin (x)\, dx \). Then you can easily see that \( du=2x\, dx \) and
\( v=-\cos (x) \). Applying parts gives \[
\int x^{2}\sin (x)dx=-x^{2}\, \cos (x)+\int 2x\cos (x)\, dx.\]
 This doesn't finish the problem, but you apply parts the second time to the
left-over integral, and it will. This time, \( u=x \) (the \( 2 \) can be
taken out of the integral) and \( dv=\cos (x)\, dx \), and so \( du=dx \)
and \( v=\sin (x) \).  Then, \[
2\int x\cos (x)\, dx=2x\, \sin (x)-2\int \sin (x)\, dx.\]
 Finally, \[
\int x^{2}\sin xdx=-x^{2}\, \cos x+2x\, \sin x+2\cos x+C.\]

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