MATH 181 Study Guide - Final Guide: Improper Integral, Irreducible Fraction, Ratio Test
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Problem 1 solution: evaluate: (a) z x2 cos(x3) dx. Solution: (b) z dx x2 5x + 4 (c) z dx. 4x2 + 1 (d) z sin5 x dx (a) we solve the integral using the u-substitution method. 3 sin u + c sin(x3) + c (b) we will evaluate the integral using partial fraction decomposition. First, we factor the denominator and then decompose the rational function into a sum of simpler rational functions. Next, we multiply the above equation by (x 1)(x 4) to get: 1 = a(x 4) + b(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 = 1 and x = 4 for simplicity. x = 1 : a(1 4) + b(1 1) = 1 a = . 1 x = 4 : a(4 4) + b(4 1) = 1 b =
