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\leftline{MATH 132, section 1 -- Calculus for Physical and Life Sciences}
\leftline{Discussion 1 -- Integration By Substitution Practice}
\leftline{January 30, 2008}
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\noindent
Each of the following integrals can be done by the method of $u$-substitution:
\begin{enumerate}
\item Determine the appropriate function $u$, 
\item Compute $du$, 
\item Change the integral to an equivalent form in the new variable $u$, 
\item Integrate in terms of $u$, then 
\item Resubstitute $u$ to express the answer in terms of the original variable.
\end{enumerate}
Do all the integrals below, and put the work for your group's integral up 
on the board.
\bigskip
\begin{enumerate}
\item[(I)] $\displaystyle \int (x+3)(x^2 + 6x + 12)^{5/4}\ dx$
\vskip 10pt
\item[(II)] $\displaystyle \int \sin(1 + \sin\theta)\cos\theta\ d\theta$
\vskip 10pt
\item[(III)] $\displaystyle \int \cos^3(4\theta) \sin(4\theta)\ d\theta$
\vskip 10pt
\item[(IV)] $\displaystyle \int \frac{1}{\sqrt{1 - 4x^2}}\ dx$
\vskip 10pt
\item[(V)] $\displaystyle \int \frac{x}{\sqrt{1 - 4x^2}}\ dx$
\vskip 10pt
\item[(VI)] $\displaystyle \int \frac{1}{x\ln(x)}\ dx$
\vskip 10pt
\item[(VII)] $\displaystyle \int \frac{\cos(\sqrt{y})}{\sqrt{y}}\ dy$.
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\item[(VIII)] $\displaystyle \int \frac{x}{\sqrt{x+1}}\ dx \qquad$ (Hint: Try $u = x+1$.)
\vskip 10pt
\item[(IX)] $\displaystyle \int \cot\theta\ d\theta \qquad$ (Hint: Rewrite $\cot\theta$
in terms of $\sin\theta$ and $\cos\theta$ to start.) 
\end{enumerate}
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