Yes, we can do this with CA. I am skiddish about doing this. Hope I don't make a mistake somewhere.

Make the sub

Then, we can write it as

Try using a 'wedge' in the 1st quadrant of the complex plane with angle

Now, using

By the residue theorem,

Because there are no singularities, the enclosed residue = 0.

Since the wedge is broken up into three different sections along the contour, we can break it up into three separate integrals. Call them a,b, and c for instance.

Take the radius of the wedge to be fixed for now at r=R and use the polar:

First, take a peek at curve a, which lies on the x-axis.

Along this curve,

Curve b, is circular in its path at radius R, so while varies, r is fixed. So, along b, .

Next, along c, is fixed. But this time

So, the integral from above becomes:

Now, let . On b, as ,

we get

because is a lot faster than .

Remember that . Then we are left with

The first one is a familiar one which equals

By Ol' Euler:

Equate real and imaginary and note that

and

Therefore, from the top,