MATH-0042 Study Guide - Midterm Guide: Trigonometric Substitution, Riemann Sum, Density
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Solutions to math 42 first exam january 28, 2014: (12 points) compute each of the following integrals, showing all of your reasoning. (a) z 2 t2et dt. 0 (6 points) let i be the integral we have to compute. We start by integrating by parts: let u = t2, dv = et dt so that du = 2t dt, v = et. Call this last integral j: we integrate by parts again, taking f = t, dg = etdt so that df = dt and g = et. 0 = 2e2 e2 + 1 = e2 + 1. Plugging this in the rst equality that we found for i, we obtain. I = 4e2 2(cid:0)e2 + 1(cid:1) = 2e2 2. (b) z sec4 2 tan4 2 d (6 points) let i be the integral we have to compute. Recall the trigonometric formula which we use to write our integral in the following form: sec2 t = 1 + tan2 t.
