MATH 181 Study Guide - Final Guide: Trigonometric Substitution
Document Summary
Solution: we will evaluate the integral using partial fraction decomposition. First, we decompose the rational function into a sum of simpler rational functions. Next, we multiply the above equation by x(x 1) to get: x(x 1) Then we plug in two di erent values for x to create a system of two equations in two unknowns (a, b). We select x = 0 and x = 1 for simplicity. 1 = a(x 1) + bx x = 0 : a(0 1) + b(0) = 1 a = 1 x = 1 : a(1 1) + b(1) = 1 b = 1. Finally, we plug these values for a and b back into the decomposition and integrate. B x 1(cid:19) dx dx =z (cid:18) a. = ln|x| + ln|x 1| + c x. Solution: the integral is computed by rewriting the integral using the pythagorean identity cos2 + sin2 = 1. Z sin2 cos3 d =z sin2 cos2 cos d .
