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\centerline{Mathematics 136 -- Advanced Placement Calculus}
\centerline{Practice with the Table of Integrals}
\centerline{October 28, 2009}
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\noindent
{\it Background and Example Problem}
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In many cases, using the table of integrals effectively involves
both
\begin{itemize}
\item Recognizing which table entry might apply from the form of the function, \emph{and}
\item Making a preliminary $u$-\emph{substitution} to put the integral you want
into the form given in the table entry.  Today we want to practice recognizing which
forms apply and doing the necessary substitutions. 
\end{itemize}
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\noindent
{\it Discussion Problems}
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\begin{enumerate}
\item[A)]  Do a preliminary substitution based on the form and integrate by
the table:
\begin{enumerate} 
\item[1)]  $\displaystyle{\int \sqrt{7 + 3e^x}\ dx}$.
\item[2)]  $\displaystyle{\int \tan^3(\ln(x))\frac{dx}{x}}$. 
\item[3)]  $\displaystyle{\int \frac{\sin x}{\cos^2 x + 6\cos x + 10}\ dx}$.
\item[]  (Hint:  Either before or after you substitute, you will also want to 
complete the square on the bottom.)
\item[4)]  $\displaystyle{\int x^{3/2} e^{x^{1/2}}\ dx}$.  
\end{enumerate}
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\item[B)]  Each of the integrals below looks very much like the others,
but they are actually quite different forms.  For each, say which method
and/or table entries you need, and do the integration:
\begin{enumerate}
\item[1)]  $\displaystyle{\int \frac{x}{\sqrt{9 - x^4}}\ dx}$
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\item[2)]  $\displaystyle{\int \frac{x}{\sqrt{9 - x^2}}\ dx}$
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\item[3)]  $\displaystyle{\int \frac{1}{x\sqrt{9 - x^2}}\ dx}$
\end{enumerate}
\end{enumerate}

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