Integrating complex numbers

I was asked to evaluate the following:

where C is the circle

Well I figure that is the function is analytical then according to Cauchy's intergral theorem:

This is somewhat complicated since it's difficult to separate the function into the real and imaginary part to take their derivatives. I pluged it into the calculator and it seems like it's not analytical. So, I went about trying to do it.

and

then it becomes:

let:

and

then:

and

And using by parts:

since r=2, and we have:

Now pluging in the limits and 0 we have:

How can this be? It shouldn't be 0 since it's not analytical. What am I missing here?