
Exercises relating to various topics we have studied.

A key skill to practice is deciding which of the several techniques we have learned should be applied to a particular problem. In these exercises and questions, you will be responsible for choosing a method which works for you.
Compute the indefinite integral. (You should choose an answer which equals $-1$ at $x = 1$; if your answer does not satisfy this, add a constant to make it so.)
Integrate by parts.
Let $u = \ln x$ and $dv = 1$.
Compute the indefinite integral. (You should choose an answer which equals $0$ at $x = 0$; if your answer does not satisfy this, add a constant to make it so.)
Partial fractions won’t help in this case because the expression is already simplified as much as possible.
Try a trigonometric substitution
Make the substitution $x = \tan \theta$.
You should arrive at the integral $\int \cos ^2 \theta \, d \theta$ after the substitution; here you’ll need to use a power-reduction formula to continue.

### Sample Quiz Questions

The region in the plane between the $x$-axis and the graph in the range $\frac {1}{5} \leq x \leq 1$ is revolved around the axis $x = \frac {1}{10}$. Compute the volume of the resulting solid.

$\displaystyle \frac {11}{25} \pi$ $\displaystyle \frac {14}{25} \pi$ $\displaystyle \frac {16}{25} \pi$ $\displaystyle \frac {19}{25} \pi$ $\displaystyle \frac {21}{25} \pi$ $\displaystyle \frac {24}{25} \pi$

Consider the region given by $2 \pi \leq x \leq \frac {5}{2} \pi$ and $0 \leq y \leq \sin {x}$. Compute the $x$-coordinate of the centroid (i.e., assuming constant density).

$\displaystyle -1 + \frac {5}{2} \pi$ $\displaystyle 1 + 2 \pi$ $\displaystyle \frac {5}{2} \pi$ $\displaystyle -1 + 3 \pi$ $\displaystyle 3 \pi$ $\displaystyle 4 \pi$

### Sample Exam Questions

An object moves in such a way that its acceleration at time $t$ seconds is $(t^2 + 5t + 6)^{-1}$ meters per second squared. If the initial velocity of the object is $2/3$ meters per second, what is the limit of its velocity as $t \rightarrow \infty$?

$\displaystyle \ln \frac {3}{2}$ meters per second $\displaystyle \ln 6$ meters per second $1$ meters per second $\displaystyle \ln \frac {4}{9}$ meters per second $\displaystyle \ln \frac {9}{4}$ meters per second $0$ meters per second

Find the volume of the solid generated by revolving the region bounded above by $y = \sin x$ and bounded below by $y = 0$ for $0 \leq x \leq \pi$ about the line $x = \pi$.

$\pi ^2$ $2\pi ^2$ $4\pi ^2$ $\displaystyle \frac {\pi ^2}{2}$ $\displaystyle \frac {\pi ^2}{4}$ none of these

Evaluate $\displaystyle \int _1^2 x \ln (x^2 + 1) dx$.

$0$ $1$ $\ln 2$ $\displaystyle \frac {1}{2}$ $\displaystyle \ln (2) - \frac {1}{2}$ none of these