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\centerline{Mathematics 136 -- Advanced Placement Calculus}
\centerline{Practice Problems on Integrals by Substitution}
\centerline{October 16, 2009}
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\noindent
Integrate each of the following by an appropriate substitution, and check your
work by differentiating your answer.  (\emph{Turn over for problem 5.}) 
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\begin{enumerate}
\item[1.]  $\displaystyle{\int \sin^7(2x) \cos(2x)\ dx.}$
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\item[2.]  $\displaystyle{\int \frac{\tan^{-1}(x)}{x^2 + 1}\ dx.}$
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\item[3.]  $\displaystyle{\int \frac{x^2 + 3}{x^3 + 9x + 14}\ dx.}$
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\item[4.]  $\displaystyle{\int \frac{e^{\sqrt{x}}}{\sqrt{x}}\ dx.}$
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\item[5.]  $\displaystyle{\int x (x + 3)^{3/5}\ dx.}$  (Let $u = x + 3$ and convert \emph{everything}
to the new variable $u$.)
\end{enumerate}
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