Complex Analysis Homework: Calculating Integral

[asi123]

Homework Statement

Hey guys.

So, I need to calculate this integral. I uploaded what I tried to do.
First of all, did the substitute, then I tried to use the residue theorem so I was looking for the residue of z=0 which is happen to be a removable singular point so it's just 0, then I went for the z=2*pi*k (when k can't be 0) residue and found out that it's 0.
I guess I have a mistake there, any idea guys?

Thanks.

Homework Equations

The Attempt at a Solution

 

[gabbagabbahey]

Your substitutions a little fishy; does e^{z}=x^2+1 really mean that x=\sqrt{e^z-1}? How are you excluding the negative root?

 

[HallsofIvy]

If you are going to use residues, what close path are you going to integrate over?

 

[asi123]

If you are going to use residues, what close path are you going to integrate over?

I thought about a closed contour consisting of the semi-circle with radius r and centre at z = 0 and the line segment going from z = -r to z = r and then doing the r --> oo thing. That way, I'll only have one pole, p=i.
Then, I'll try to break it into -r to 0 and from 0 to r, and find the latter, does that seems right?

Thanks.