This is what I did... I'm not sure if I'm in the right direction about this problem.
I am french so excuse me if my translation in english isn't accurate.
I used the residual theorem.
The integral of is 0. (Cauchy's Theorem).
Poles of order 1 at
Only the pole at z = ia is in the semi-circle drawn above.
Residual at z = ia :
What does cos(ia) represent ? Does that even exist?
If this is totally wrong, then how do I solve this problem?
where is the upper half circle contour. I think it's not to hard to show:
where is the half-arc contour and on the straight-line segment along the real axis we have:
(since the part over C is zero).
 I made a silly mistake on this initially but corrected it here. Hope that didn't cause problems for you Illusion.