I expect you would have to use the residue theorem...
A solution of the integral based on the Laplace Tranform is illustrated here...
http://www.mathhelpforum.com/math-he...rm-170211.html
Kind regards
Here is a third method using the Laplace transform
Consider the function
Note that
Is the integral you are looking for
Using a trig Identity we get
Now if we take the Laplace transform with respect to t we get
Now by partial fractions on the 2nd term we get
Plugging this back in gives
Now finally take the inverse transform to get