I've been trying to Fourier expand
$$\psi(t)=\sin\left(2\pi at\ +\ \sin\left(2\pi bt\right)\right)$$
Please help me evaluate the integral:
$$\nu(x)=\frac{1}{\sqrt{ab}}\int_0^{ab}\sin\left(2\pi at+\sin\left(2\pi bt\right)\right)\cdot e^{-\frac{2\pi ixt}{\sqrt{ab}}}dt$$
(a,b being constants - frequency with my case)
(please consider the fact that I'm not a complex analysis guy, actually a High Schooler, and that I cannot fathom your answer if it has all advanced terminologies [I know what Lebesgue Space is though] )
Edit/Bump: I got till here:
$$\nu(x)=\int_0^{\frac 12}\cos (\omega_at+m\sin \omega_bt)\cdot \cos (xt)dt$$
I'm able to numerically get the Spectrum using Desmos and also taking reference from a Video on MIT Opencourseware on Signals Processing but not mathematically perform this integral.
My attempt
I think the methods I've tried all along evolved with time. I'm facing a problem with the integration part. I first tried u-sub, tried trigonometric simplification and all of them didn't work. Back then I did not realize that it was periodic over $ab$ and that I shouldn't have attempted a Fourier Transform. Then I realized that it was periodic, applied Integration by parts. Didn't work. I tried doing the "Feynmann trick of partial inside the integral sign", I got till a differential equation to which I got an exponential function as a non-constant solution. I found it to be a very weird solution and I realized that the Fourier Coefficients are time-dependent and my differential equation doesn't work.
Using Transformation Formulae gives 2 terms that are 3 trig-functions multiplied. Applying more Transformation helps, I think it leads us back to the original form?