MATH 2B

Midterm 2 | Winter '17 | Youssefpour

1. True or False (you do NOT need to provide any explanation)
a) The integral $\int_1^x \frac{1}{x^3} dx$ is convergent.

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b) If $f(x) \ge g(x) \ge 0$ and $\int_a^\infty g(x) dx$ diverges, then $\int_a^\infty f(x) dx$ also diverges.

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c) If $f(x) \ge g(x) \ge 0$ and $\int_a^\infty f(x) dx$ diverges, then $\int_a^\infty g(x) dx$ also diverges.

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d) $\frac{x^2 - 4}{x(x^2 + 4)}$ can be put in the form $\frac{A}{x} + \frac{B}{x^2 + 4}$.

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e) $\frac{d}{dx} \int_0^{\pi/4} \sin(x) dx) = 1 - \frac{\sqrt{2}}{2}$.

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2. Evaluate the following indefinite integrals.
a) $\int \frac{\sqrt{x^2 - 4}}{x^4} dx$

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b) $\int \frac{x^2}{e^{-2x}} dx$

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c) $\int 4\tan^3(t) \sec^5(t) dt$

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d) $\int \sin u \sqrt{1 + \cos u} du$

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3. Evaluate the following definite integrals.
$\int_0^1 \tan^{-1}(t) dt$ (Hint: Remember $\tan^{-1}(t) = \arctan(t)$)

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4. Find the average value of the function on the given interval.
$f(x) = \cos 4x, [0, \frac{\pi}{2}]$

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5. Find the exact length of the curve $y = \ln (\cos x), 0 \le x \le \frac{\pi}{3}$

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6. Evaluate the following improper integral and show whether it is convergent or divergent.
$f(x) = \sin 4x, [0, \frac{\pi}{4}]$

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