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Fall 2006 - Takeda's Class - Final Exam

Fall 2006 - Takeda's Class - Final Exam - Math 20B Final...

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Math 20B, Final Exam Solutions December 7, 2006 Name: Section: This exam consists of 11 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 two 4-by-6 index cards, both sides. 4. You have two hours for this exam. Score 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 Total 100 1

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1. (a) Evaluate 2 1 x ln( x 2 ) dx Let u = x 2 . Then du = 2 xdx and x 1 2 u 1 4 Then 2 1 x ln( x 2 ) dx = 4 1 1 2 ln u du = 1 2 ( u ln u - u ) 4 1 = 1 2 [(4 ln 4 - 4) - ( - 1)] = 2 ln 2 - 3 2 . (b) Find the indefinite integral xe x dx By integration by parts, xe x dx = xe x - e x + C. 2
2. Compute 2 1 1 x - 1 dx . (Note: The integrand is not defined at x = 1.) Since 1 x - 1 is not defined at x = 1, this is an improper integral. Thus 2 1 1 x - 1 dx = lim t 1 + 2 t 1 x - 1 dx = lim t 1 + 2 x - 1 2 t = lim t 1 + (2 - 2 t - 1) = 2 . 3

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3. Evaluate ln( x 2 + 1) dx . (Hint: Remember how to compute ln x dx .) Let du = dx and v = ln( x 2 + 1). Then u = x and dv = 2 x x 2 +1 dx . Thus ln( x 2
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