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\Large
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Exam 2 \hspace{40mm}Name: \underline{\hspace{30mm}}

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\small
Directions:  Answer each question to the best of your ability.  The point value of each question follows the question.  Calculators are allowed, but you must show appropriate work to receive full credit.  
Choose the best response for the following 5 questions. (4 pts each)

\begin{enumerate}
\item Which is the correct use of substitution for the definite integral, $\int_{_{\frac{\pi}{2}}}^{^{\pi}} \, \cos^2 x \sin x \,dx$
\begin{enumerate}
\item $-\int_{_{\frac{\pi}{2}}}^{^{\pi}} \, u^2\,du$
\item $\int_{_{\frac{\pi}{2}}}^{^{\pi}} \,u^2\,du$
\item $-\int_{_0}^{^{-1}} \,u^2\,du$
\item $\int_{_0}^{^{-1}} \, u^2\, du$

\end{enumerate}

\item How many applications of the integration by parts method are necessary to solve $ \int x^4\, \sin x\, dx$?
\begin{enumerate}
\item 2
\item 4
\item 6
\item This integral cannot be solved by the integration by parts method
\end{enumerate}

\item Which approximation always give an overestimate of $ \int^{^b}_{_{a}} f(x) \, dx$ when $f''(x)<0$
\begin{enumerate}
\item Left-hand Rule
\item Right-hand Rule
\item Midpoint Rule
\item Trapezoid Rule
\end{enumerate}

\item Given that   \[ \lim_{x \rightarrow \infty} f(x)=0 \]  and $a$ is a constant, $ \int_a^{\infty}\,f(x)\, dx$:
\begin{enumerate}
\item Always diverges.
\item Always converges.
\item Convergence or divergence depends on $f(x)$.
\item No determination regarding divergence or convergence can be made.
\end{enumerate}

\item Approximating an integral with 8 subdivisions gives an error of 5. Approximating the same integral with the same rule with 80 subdivisions gives an error of .05.  Which approximation rule was used?
\begin{enumerate}
\item Left-hand Rule
\item Right-hand Rule
\item Midpoint Rule
\item Simpson's Rule
\end{enumerate}
\newpage
\item Following are methods of finding integral that we have studied along with the necessary information we have to identify.  Given the integrals, find the appropriate method of integration and identify the relevant information for that method.  use numerical only when all other options fail. Assume $a$ and $b$ are constants.

\vspace{5mm}






\begin{tabular}{|c|c|}
\hline
Method&Information to Identify\\
\hline
Elementary integral&None (we can take integral directly)\\
\hline
Substitution&O.F.=$f(u)$;  I.F. = $u=f(x)$;   $du=f'(x)dx$\\
\hline
Integration by Parts& $u=$ \hspace{10mm}   $v'=$\hspace{10mm}   $u'=$\hspace{10mm}   $v=$\\
\hline
Integration by Partial Fractions&Decompose fraction into sum \\
\hline
Numerical Approximation&None (must use one of 5 numerical methods)\\
\hline

\end{tabular}

\vspace{10mm}

\begin{tabular}{ccc}
Integral & \hspace{7mm} Method (2 pts each) \hspace{2mm} &Necessary Information for the method (3 pts each) \hspace{5mm} \\[10 pt]


$ \int_a^b \,  x^5 e^x\, dx$&&\\[60 pt]


$ \int_a^b \, x^3 e^{x^4} \, dx$&&\\[60 pt]


$ \int_a^b \, \frac{1}{1+x^2} \, dx$&&\\[60 pt]


$ \int_a^b \, \frac{\sin^2 x}{x^2} \, dx$&&\\[60 pt]


$ \int_a^b \, \frac{2x+1}{x^2-2x-3} \, dx$&&\\[60 pt]

\end{tabular}

\newpage
Find each of the following integrals.  Don't forget to show all work.  Clearly identify the integration method used and necessary information.  Please keep your work neat and organized.  You may choose to not do one of the three problems 7, 8, or 9.  You may do all three for possible 5 pts. extra credit.  (5 pts each)



\item $ \int\, x^2e^{2x^3-5}\, dx$
\vfill
\item $ \int \, e^x \cos 4e^x \, dx$
\vfill
\item $\int \, \frac{x+5}{x^2+10x-11} \, dx$
\vfill
\newpage
\item $ \int \, x^8 \ln x \, dx$
\vfill
\item  $\int \frac{3x+8}{x^2+7x+6}$
\vfill
\newpage
\item Identify each of the following integrals as either improper or not improper.  (1 pt each)
\begin{enumerate}
\item $\int_{1}^4\, \frac{1}{x^4}\,dx$
\item $ \int_{-1}^1\, \frac{1}{x^3+1}\,dx$
\item $\int_{1}^{\infty}\, \frac{1}{x}\,dx$
\item $\int_0^7\, \frac{1}{x^2-4}\, dx$
\item $\int_{-1}^2\, \frac{1}{x+2}\,dx$
\end{enumerate}
\item Determine if the following integrals converge or diverge.  If it converges, find the value.  Again, don't forget to show your work using limits.  (5 pts each)
\begin{enumerate}
\item $ \int_0^1\,\frac{2}{ \sqrt[4]{x}  }\, dx$
\vfill
\item $ \int_2^{\infty}\,\frac{5}{\sqrt[3]{x}}\,dx$
\vfill

\end{enumerate}
\newpage
\item An object moves with velocity, $v(t)=\frac{1}{5}t^3e^t$
\begin{enumerate}
\item Complete the table below (1 pt each)

\begin{tabular}{|c|c|c|c|c|c|}
\hline
$t$& \hspace{5mm} 0 \hspace{5mm} & \hspace{5mm} 1 \hspace{5mm} & \hspace{5mm} 2 \hspace{5mm} & \hspace{5mm} 3 \hspace{5mm} & \hspace{5mm} 4 \hspace{5mm} \\
\hline
$v(t)$&&&&&\\
\hline
\end{tabular}

\end{enumerate}
Estimate the total distance traveled from 0 to 4 using the following rules with the number of subdivisions in parentheses. Remember, distance = $ \int_0^4\,v(t)\,dt$

\begin{enumerate}
\item Trapezoid Rule(2) (5 pts)
\vfill
\item Midpoint Rule(2) (5 pts)
\vfill
\item Simpson's Rule(4) (5 pts)
\vfill

\end{enumerate}
\end{enumerate}
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