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


 

Kevin B Mcleod
3/4/2002