Would've been so much nicer if it was a cos instead of a sin lol.
Been a while since I've done one of these, but I'm guessing: take the CCW quarter-circle contour C around the first quadrant with radius R approaching infinity, so that:
where C1 is the quarter circle arc (from z = R to z = iR), C2 goes from iR to 0, and C3 goes from 0 to R?
The contour over C can then be evaluated using Residue theorem, with with z = +i being the only singularity within C.
The contours over C1 and C2 can be evaluted by using an appropriate parameterisation (alternatively, I'm thinking that the integral over C1 just be 0 since the denominator goes as R^2?).
The original integral can then be extracted (or at least I think it can...) by taking imaginary parts of both sides, so that
Is this correct? I'm a bit hesitant on that last step, not entirely convinced myself if that last expression holds.
I don't normally spend time thinking about integration questions, so that's enough integration for me for the next year haha.