MATH 31B Study Guide - Midterm Guide: Trigonometric Substitution
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Winter 2013: compute the inde nite integral (cid:90) x + 1 x2 + 16 dx. We decompose the integral into two parts: (cid:90) (cid:90) (cid:90) x + 1 x2 + 16 dx = x x2 + 16 dx + For the rst summand we use the substitution x2 = u (so 2xdx = du) to get x x2 + 16 dx = ln|u + 16| + c = For the second summand we use the trigonometric substitution x = 4 tan (so dx = (4 sec2 )d ) to get (cid:90) (cid:90) 16(tan2 + 1) (cid:90) (4 sec2 )d . Therefore, the original integral equals ln|x2 + 16| + + c: this problem asks you to set up the numerical approximations for the integral (cid:90) 1. You don"t have to compute the answer, just leave it as a sum. For example, for the midpoint approximation m3, it su ces to write:
