This document presents a method for evaluating integrals of the form I=x^a R(x) dx, where R(x) is a rational function with at most a simple pole at the origin and 0<a<1. It involves changing variables to write the integral in terms of a contour integral over a keyhole contour C. By taking limits, the contour integral can be written as a sum of integrals over the real segments of C. Taking another limit then allows writing the original integral I in terms of the residues of the rational function R(z). The final result obtained is that I=2πi{Σ Res z^a R(z)}/(1-2a), where the sum is over