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\begin{center}
{\Large Warm-Up Problems and Lecture Problems

February 7, 2003}
\end{center}


\begin{enumerate}

\item For each of the following integrals, determine the correct choice of trigonometric substitution.  (Use the chart handed out last time.)
	\begin{enumerate}
	\item  \[ \int \frac{1}{x^2\sqrt{x^2-25}} \; dx \]
		\bigskip
	\item  \[ \int \frac{1}{(25+16x^2)^{3/2}} \; dx \]
		\bigskip
	\item  \[ \int \frac{\sqrt{5-x^2}}{x} \; dx \]
		\bigskip
	\end{enumerate}

\item  For the substitutions made in the previous problem, convert the following expressions into expressions of $x$.  For example, if your substitution was $x=\sin \theta$ and you are asked to put $\sin^2\theta$ into an expression of $x$, you would write $\sin^2\theta = x^2$.

Remember to draw the appropriate triangle!
	\begin{enumerate}
	\item  $\theta+\cos \theta -\tan \theta$
		\bigskip
	\item  $\theta+\cos \theta -\tan \theta$
		\bigskip
	\item  $\theta+\cos \theta -\tan \theta$
		\bigskip
	\end{enumerate}

\newpage

\item  Write the following fractions in the correct ``partial fraction decomposition form:''
	\begin{enumerate}
	\item  \[ \frac{x^2+3x}{(x^2+x+1)(x-2)} \]
		\bigskip
	\item  \[ \frac{5x^3-1}{(2x+3)^3(x+4)} \]
		\bigskip
	\end{enumerate}

\newpage

\item  Solve for the constants in the following partial fraction decomposition:
	\[
	\frac{5}{(2x+1)(x-2)} = \frac{A}{2x+1} + \frac{B}{x-2}
	\]

\newpage

\item  Here is an integral that I have done the partial fraction decomposition.
Do the rest of the work and compute the antiderivative.
(You might want to perform the partial fraction decomposition at home later.)
\[
\int \frac{5x^3-3x^2+2x-1}{x^4+x^2} \; dx 
	= \int \left( \frac{2}{x} - \frac{1}{x^2} + \frac{3x-2}{x^2+1} \right) \; dx 
\]



\end{enumerate}

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