MATH 181 Study Guide - Final Guide: Antiderivative
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Problem 1 solution: a) i) find an antiderivative for the function f (x) = x cos x, compute the de nite integral , compute the de nite integral ! 0 x cos x dx: i) find an antiderivative for the function f (x) = xex. xex dx. Solution: a) i) an antiderivative for f is a function f such that: F (x) = ! f (x) dx = ! x cos x dx. Let u = x and v = cos x. Using the integration by parts formula: we get: uv dx = uv ! u v dx. F (x) = ! x cos x dx = x sin x ! sin x dx. 0 x cos x dx = f ( ) f (0) = ( sin + cos ) (0 sin 0 + cos 0) = (0 1) (0 + 1) = 2: i) an antiderivative for f is a function f such that:
