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\title{Math 1b. Integration by Substitution
}

\date{Spring 2006}

\begin{document}

\maketitle



\begin{enumerate}

\item 
Evaluate each of the following integrals
\begin{enumerate}

\item 
$\displaystyle \int \cos(2x + 3) \, dx$

\vfill

\item 
$\displaystyle \int (3x-2)^4 \, dx$

\vfill

\item 
$\displaystyle \int \frac{2 x^3}{1 + x^4} \, dx$

\vfill

\item 
$\displaystyle \int  \frac{dx}{1 + \sqrt{x}}$.

\vfill

\clearpage

\item 
$\displaystyle \int \frac{\arcsin x}{\sqrt{1 - x^2}} \, dx$

\vfill

\item 
$\displaystyle \int \frac{e^x}{(2e^x + 3)^2} \, dx$

\vfill

\item 
$\displaystyle \int \sin^3 x \, dx$

\vfill

\item 
$\displaystyle \int_0^{\pi} \sin^3 \theta \cos \theta \, d\theta$

\vfill

\end{enumerate}

\item
Find real numbers $a$ and $b$ such that the equality
$$
\int_1^2 x^2 e^{x^3/4} \, dx = \frac{4}{3} \int_a^b e^u \, du
$$
holds.

\vfill

\item
Suppose that $\int_0^{12} g(x) \, dx = \pi$.  Evaluate $\int_0^4 g(3x) \, dx$.

\vfill

\end{enumerate}


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