I'm trying to crack this problem but I can't seem to get it right.

It's the third exercise from from chapter 2 of "Complex Analysis" by Stein & Shakarchi.

I've drawn the sector (contour C) here (sorry for the crappy MS-paint picture):

As there are no poles within the sector I set , so that leaves

Now, I assume one of these three components goes to zero as A goes to infinity and indeed the second one (with as variable) does. Then I'm stuck because what I find doesn't lead me to anything remotely like the answer Wolfram's online calculator gave me if I use integration by parts

Can anyone help me with this problem, what am I doing wrong?