MATH 31B Study Guide - Midterm Guide: Direct Comparison Test, Partial Fraction Decomposition, Improper Integral
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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 . Problem 1: (a) evaluate the inde nite integral: z tan2 x sec6 x dx (b) evaluate the inde nite integral: z arctan x (1 + x2)3/2 dx. Solution: (a) use the identity sec2 x = (1 + tan2 x) and then substitute u = tan x: Z tan2 x sec6 x dx = z tan2 x(1 + tan2 x)2 sec2 x dx = z u2(1 + u2)2 du u7. + c. (b) make the subsitute x = tan , dx = sec2 d : Z arctan x (1 + x2)3/2 dx = z (1 + tan2 )3/2 sec2 d = z cos d .
