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Math 132: Calculus for the Physical \& Life Sciences 2

Problem Set 4

Due Friday, February 15, 2008, at the beginning of class.
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\noindent
General Directions:  You must show all work for credit on these
problems.

\begin{enumerate}
\item Use trigonometric substitutions (possibly after completing
the square) to evaluate each of the following
integrals.

\begin{enumerate}
\item[a.]  ${\displaystyle \int \frac{1}{\sqrt{9 + 4x^2}}\ dx}$

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\item[b.]  ${\displaystyle \int \frac{\sqrt{x^2 - 1}}{x^2}\ dx}$

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\item[c.]  ${\displaystyle \int \frac{1}{\sqrt{3 - 2x - x^2}}\ dx}$

\end{enumerate}

\item  Integrate using the partial fraction technique (divide the 
denominator polynomial into the top first when necessary).

\begin{enumerate}
\item[a.]  ${\displaystyle \int \frac{1}{x^2 + x - 6}\ dx}$

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\item[b.]  ${\displaystyle \int \frac{x^4}{x^2 + 4}\ dx}$

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\item[c.]  ${\displaystyle \int \frac{x + 10}{2x^2 + 5x - 3}\ dx}$

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\item[d.]  ${\displaystyle \int \frac{x^2+1}{x^3 + 2x^2 + x}\ dx}$

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\item[e.]  ${\displaystyle \int \frac{x + 4}{x^3 + 4x}\ dx}$
\end{enumerate}

\item  Integrate using the table of integrals in Stewart, and preliminary
substitutions or algebraic manipulation as necessary.  In each case,
state clearly the number of the table entry you are using.

\begin{enumerate}
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\item[a.]   ${\displaystyle \int \frac{x^2}{\sqrt{x^2 + 25}}\ dx}$

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\item[b.]   ${\displaystyle \int \sqrt{2x - x^2}\ dx}$

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\item[c.]   ${\displaystyle \int \frac{1}{\sqrt{e^{2x} - 1}}\, dx}$

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\item[d.]   ${\displaystyle \int \frac{\cos(x)}{\sqrt{4 - \sin^2(x)}}\ dx}$

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\item[e.]   ${\displaystyle \int \frac{\sec^2(x)}{\tan^2(x) + 2\tan(x) + 2}\ dx}$

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\item[f.]   ${\displaystyle \int e^x \sqrt{1 + e^{2x}}\ dx}$


\end{enumerate}
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