600-232 CALCULUS II SAMPLE GATEWAY TEST #2

1.

(10 points) Use the method of parts to find the antiderivative ${\displaystyle \int x e^{-12x}\,dx}$.

2.

(10 points each) Find each of the following:

(a)

${\displaystyle \int \frac{1}{\sqrt{7 - x^{2}} } \,dx}$

(b)

${\displaystyle \int \cos^{2} (9\theta) \,d\theta}$

(c)

${\displaystyle \int \sin^{6} \theta \cos \theta \,d\theta}$

3.

(10 points) Write down the form of the partial fraction decomposition of
${\displaystyle \frac{x^3 + 1}{(5x+7)^{2} (5x^{2} + 7)^{3} }. }$ (Do not attempt to evaluate the coefficients.)

4.

(10 points) Given that ${\displaystyle \frac{14}{(x + 3)(2x - 1)} = \frac{A}{x + 3} + \frac{B}{2x - 1} }$, find A and B.

5.

(10 points) If $\theta$ is an acute angle and $\cos \theta = x/5$, what is $\tan \theta$?

6.

(10 points) Use a substitution to reduce ${\displaystyle \int \frac{1}{x(3\sqrt{x} + \sqrt[3]{x})} \,dx}$ to the anti-derivative of a rational function. You should carry out the substitution completely, but you should not go on and evaluate the anti-derivative.

7.

(10 points each) Use a substitution to reduce each of the following anti-derivatives to an anti-derivative involving powers of trigonometric functions. Do not attempt to go on and evaluate the anti-derivative.

(a)

${\displaystyle \int x^8 \sqrt{4 - 9x^2} \,dx }$

(b)

${\displaystyle \int \frac{x^6}{\sqrt{9x^2 - 4}} \,dx }$