I'm looking to solve the inhomogeneous wave equation $$ u_{tt} - c u_{xx} = -A q \sin(\omega t -q x)$$ where $\omega q = c$. In order to do that, as I have read after some searching, I should find the solution in the form of a specific solution and the general homogeneous solution. $$ u = u_0 + u_p $$ The solution to the homogeneous equation, if my understanding is correct, is in its simplest form: $$ u_0 = A e^{i(kx - \omega t)} + B e^{-i(kx + \omega t)} $$ where $A$ and $B$ $\in \mathbb{C}$.
Now for the specific solution which is actually the core of my question, I have read that the solution is of the form (for my specific case where the source is the one written in the first equation):
$$ \int_0^t \int_{x-c(t-s)}^{x+c(t-s)} -Aq \sin(\omega s -qy) \,\mathrm{ds}\,\mathrm{dy} $$
I can't quite understand how should I solve this. I've tried integrating first along dy and then with respect to ds, but I'm not so certain that doing so is correct. Furthermore, asking WolframAlpha has given me an answer which is different from what I found, and it also depends on which order is the integration taken.
If you have an insight about this problem, and if you find some mistake within this development (I mean with the basics of what I have written up until the double integral), please let me know. Thank you for your help!