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31B midterm2-sols

31B midterm2-sols - MATH 31B LECTURE 4 MIDTERM 2 Name...

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MATH 31B, LECTURE 4 MIDTERM 2 FEBRUARY 27, 2012 Name: Solutions UID: TA: (circle one) Huiyi Hu Brent Nelson Andrew Ruf Discussion meets: (circle one) Tuesday Thursday Instructions: The exam is closed-book, closed-notes. Calculators are not permitted. Answer each question in the space provided. If the question is in several parts, carefully label the answer to each part. Do all of your work on the examination paper; scratch paper is not permitted. If you continue a problem on the back of the page, please write “continued on back”. Each problem is worth 20 points. Problem Score 1 2 3 4 5 Total 1
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2 Problem 1: (a) Evaluate the indefinite integral: Z tan 2 x sec 6 x dx (b) Evaluate the indefinite integral: Z arctan x (1 + x 2 ) 3 / 2 dx . Solution: (a) Use the identity sec 2 x = (1 + tan 2 x ) and then substitute u = tan x : Z tan 2 x sec 6 x dx = Z tan 2 x (1 + tan 2 x ) 2 sec 2 x dx = Z u 2 (1 + u 2 ) 2 du = u 7 7 + 2 u 5 5 + u 3 3 + C = tan 7 x 7 + 2 tan 5 x 5 + tan 3 x 3 + C. (b) Make the subsitute x = tan θ , dx = sec 2 θ dθ : Z arctan x (1 + x 2 ) 3 / 2 dx = Z θ (1 + tan 2 θ ) 3 / 2 sec 2 θ dθ = Z θ · cos θ dθ
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