Calculation of coefficients of a Fourier series

Question

Calculating the Fourier series of a periodic function I need to evaluate these integrals: $$1) \int_{-\pi}^{\pi}dt\left(\cos^{-1}(\alpha t-1)+2(1-\alpha t)\sqrt{\frac{1}{2}\alpha t-\frac{1}{4}\alpha^2t^2}\right)\sin(t)$$ $$2) \int_{-\pi}^{\pi}dt\left(\cos^{-1}(\alpha t-1)+2(1-\alpha t)\sqrt{\frac{1}{2}\alpha t-\frac{1}{4}\alpha^2t^2}\right)\cos(t)$$ Knowing alpha, it's easy to calculate $1)$ and $2)$ numerically. The problem is that I don't know $\alpha$, so I would like to have an analytical evaluation of these two integrals. Is there some method to calculate them? Thanks.

Answer 1

Probably not too helpful (kind of obvious), but I think the first step is to recognize that each of them is an integral of a sum, which can be simplified to a sum of integrals.

As for actually integrating the part on the right....not sure.