Part (a) asks to compute the integral by means of the residue at x = -1. I have done this just now, and the answer is $\pi$.
Part (b) asks, "can you see a simpler way to do it? explain."
I tried ordinary calculus methods, making two substitutions: once for the square root, and once for $tan(\theta)$, and get 1 in the integrand, after the trigonometric functions $\sec^2(\theta)$ cancel each other out. But this integral diverges, obviously.
So, what is the simpler way to do it, if it's not by ordinary calculus? Would it perhaps be another complex analysis method that's not so apparent?