I've been working on a similar problem myself, i've plotted the graph in MATLAB and see that it is an odd function, equating all A terms to 0.
I'm having trouble doing the integral of t*exp(-abs(t))sin(wt) thought. Any help would be greatly appreciated!!
Split the integral into two parts:
in the second integral change the variable :
Now the second integral is minus the complex conjugate of the first so we can write:
and the integral inside the rightmost brackets can be done by parts.