I'm interested in evaluating the following integral
$$\mathcal{J} = \int_{0}^{\infty}\frac{\sum_{k=1}^{\infty}k\sin(kx)\,e^{-tk^2}}{\sum_{k=1}^{\infty}\cos(kx)\,e^{-tk^2}} \, \mathrm{d}t$$
I think I've seen this before somewhere. Inverse Fourier? However, I do not remember where I've seen it and I do not have any starting point to begin cracking it.
The answer is supposed to be $\displaystyle \frac{\pi^2 ( \pi - x )}{8}$ for $0<x<2\pi$.