Solving an Integral Using Residue Theorem

[asi123]

Homework Statement

Hey guys.
So I got this integral I need to solve, of curse using the residue theorem.
The thing is, that I don't understand the curve.
I know that whenever Z^2 = integer, this function has a singularity point because e^(2*pi*i*n) = 1.
But again, I'm not sure what this curve has enclosed in.

Thanks.

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

This is, actually, an "infinite sequence" of problems! Each path is a circle, with center (0,0) of radius R which lies between $\sqrt{n}$ and $\sqrt{n+1}$ for each positive integer n. I suspect that you will find that the number of poles inside each path depends on n.