\subsection{\bf Criteria for Choosing Manipulation}

\begin{slide}

\begin{itemize}
\item In each case, one must simply use criterion.

\begin{description}
\item{$\bullet$} Is $f(x)$ or $f(y)$ better approximated by a polynomial?

\item{$\bullet$} One can use same criterion both to choose $g(x)$ and/or
subtract off some particularly nasty part (e.g., a singularity)
of $f(x)$.  
\end{description}

\item For example
\begin{eqnarray*}
I & = & \int^b_a f(x)\,g(x)\,dx \\
& = & I_1 + I_2 \\
\mbox{where}\ f(x) & = & f_1(x) + f_2(x) \\
\mbox{and}\ I_i & = & \int^b_a f_i(x)\,g(x)\,dx \\
\end{eqnarray*}

\item where $f_1(x)$ may be well approximated by a polynomial and
$I_1 = \int^b_a f_1(x)\,g(x)\,dx$ could be done numerically.

\item $f_2(x)$ may be a nasty function to integrate numerically
(e.g., $x^\alpha$ with $\alpha$ non-integer) but chosen so that we
can do $I_2 = \int^b_a f_2(x)\,g(x)\,dx$ analytically.
\end{itemize}
\end{slide}

