%% Math 132, Spring, 2004
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\begin{center}
\textbf{College of the Holy Cross, Spring Semester, 2005} \\
\textbf{Math 132 Review Sheet for Midterm 1}
\end{center}

\noindent The first midterm will be held the evening of Wednesday,
February~23, from 6:00--7:30. Material from Section~5.4 (properties of
the integral) through Section~7.4 (partial fractions and trig
substitution) will be covered.

\begin{enumerate}
  
\item At $t$~seconds after liftoff, a rocket experiences acceleration
  $a=32+0.32t~\text{ft/sec}^2$.

  \begin{itemize}
  \item{(a)} How long does it take the rocket to break the
  sound barrier? (Assume sound travels 1000~ft/sec.)
  
  \item{(b)} If the rocket burns for 90~seconds, how far has the
  rocket traveled?

  \end{itemize}

\item Determine whether or not the following are meaningful. Evaluate
  those that are, and explain what's wrong with those that aren't.
  \[
  \int u^2\,dx\qquad \int u^2\qquad 
  \int\frac{d\,\mathrm{cabin}}{\mathrm{cabin}}\qquad
  \int\frac{1}{x}\qquad \int d\theta\qquad \int\frac{e^u}{2x}\,dx
  \]

\item A graph $y=f(t)$ is shown. Sketch the graph of $F(x)=\displaystyle\int_1^x
  f(t)\,dt$ in the grid provided, marking the coordinates of all
  critical points.
  \begin{center}
    \input{review1a.eepic}
    \input{review1b.eepic}
  \end{center}

\item Find $\displaystyle\frac{d}{dx}\int_1^x(1+t^2)^{10}\,dt$. (Hint: Do not
  evaluate the integral!)

\item Use the substitution $u=e^t$ to transform 
  $\displaystyle\int_0^x e^{e^t}\,dt$. Be sure to handle the limits.
 
\item Find a formula for $F(x)=\displaystyle\int_0^{x}2te^{-t^2}\,dt$, then use
  your formula to compute~$F'(x)$. What happens if you change the
  lower limit of integration in the definition of~$F$?

\item Without using your notes, find formulas for the given integrals:
  \[
  \int \frac{dx}{a+x},\qquad
  \int \frac{dx}{a-x},\qquad
  \int \Bigl(\frac{1}{a+x}+\frac{1}{a-x}\Bigr)\,dx,\qquad
  \int \Bigl(\frac{1}{a+x}-\frac{1}{a-x}\Bigr)\,dx.
  \]
  Simplify your answers if possible. For the last two, use algebra to
  put the integrand over a common denominator. One of the resulting
  integrals can be done using substitution. Do this, and verify that
  you get the same answer either way.

\item Evaluate the following integrals. In each case, use a right
  triangle to simplify the integrand before you integrate.
  \[
  \int_0^x \tan(\arcsin t)\,dt,\qquad
  \int_0^x \sin(\arctan t)\,dt.
  \]

\item Use the identity
  $\cos^2\theta=\frac{1}{2}\bigl(1+\cos(2\theta)\bigr)$ to evaluate
  $\displaystyle\int\cos^2\theta\,d\theta$~and  $\displaystyle\int\cos^4\theta\,d\theta$.

\item Use the previous question to compute
  $\displaystyle\int_0^\pi\cos^2\theta\,d\theta$. Explain your result
  geometrically.

\item Use substitution and/or integration by parts to compute
  \[
  \int_5^8 x\sqrt{9-x}\,dx\qquad
  \int\frac{\sin\theta}{\cos^2\theta}\,d\theta\qquad
  \int x^5\sqrt{2-x^3}\,dx\qquad
  \int_0^1 2x\arctan x\,dx
  \]

\item Perform necessary algebra (expansion, polynomial division,
  completion of the square, partial fractions) on the following, then
  compute the integrals.
  \[
  \int \frac{x^4+1}{x^2+1}\,dx\qquad
  \int \frac{x^3-2x^2+x}{x^2-3x+2}\,dx\qquad
  \int \frac{x+1}{x^2+6x+13}\,dx\qquad
  \int \frac{2x-4}{(x^2-4x+5)^3}\,dx
  \]

\item Find the exact area under the first arch of~$f(x)=x^2\sin x$.

\item Use integration by parts to compute $\displaystyle\int v\arcsin(v)\,dv$

\item Evaluate \emph{two} of the following:
  \[
  \int \frac{1}{\sqrt{1+x^2}}\,dx,\qquad
  \int \frac{x}{\sqrt{1+x^2}}\,dx,\qquad
  \int \frac{x^2}{\sqrt{1+x^2}}\,dx,\qquad
  \int \frac{x^3}{\sqrt{1+x^2}}\,dx.
  \]

\end{enumerate}

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