Fall 2006 - Exam 2

# Fall 2006 - Exam 2 - Math 20B(Calculus for Science and...

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Math 20B(Calculus for Science and Engineering) Midterm 2, Solutions November 15, 2006 Name: Section: This exam consists of 7 pages including this front page. Ground Rules 1. No calculator is allowed. 2. Show your work for every problem. A correct answer without any justiﬁcation will receive no credit. 3. You may use one 4-by-6 index card, both sides. Score 1 10 2 10 3 10 4 10 5 10 6 10 Total 60 1

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1. Evaluate Z x 2 ln xdx Let u = ln x and dv = x 2 dv . Then du = 1 x dx and v = 1 3 x 3 . Thus Z x 2 ln xdx = 1 3 x 3 ln x - Z 1 3 x 3 1 x dx = 1 3 x 3 ln x - Z 1 3 x 2 dx = 1 3 x 3 ln x - 1 9 x 3 + C 2
2. Compute the integral Z x sin xdx by using complex exponentials. Since sin x = 1 2 i ( e ix - e - 2 i ) , we have Z x sin xdx = Z 1 2 i ( xe ix - xe - ix ) dx = 1 2 i ( 1 i xe ix + e ix + 1 i xe - ix - e - ix ) + C = - 1 2 x ( e ix + e - ix ) + 1 2 i ( e ix - e - ix ) + C = - x cos x + sin x + C. 3

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3. Find Z 1 sin x dx. (Hint: This is an integral of the type
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Fall 2006 - Exam 2 - Math 20B(Calculus for Science and...

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