MATH 181 Study Guide - Final Guide: Trigonometric Substitution, Improper Integral
Document Summary
Solution: we will evaluate the rst integral using integration by parts. Let u = arctan x and v = x. Z uv dx = uv z u v dx. Use the substitution w = x2 + 1 to evaluate the integral on the right hand side. Then dw = 2x dx dw = x dx and we get: = x arctan x ln(x2 + 1) + c. Note that the absolute value signs aren"t needed because x2 + 1 > 0 for all x. We will evaluate the second 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(x2 + 1) to get: 1 = a(x2 + 1) + (bx + c)x. Then we plug in three di erent values for x to create a system of three equations in three unknowns (a, b, c).
