I would like to compute:
$$ \int_{0}^{2\pi}\frac{\sin(nx)}{\sin(x)} \mathrm dx $$
We have:
$$ \int_{0}^{2\pi}\frac{\sin(nx)}{\sin(x)}\mathrm dx=2\int_{0}^{\pi}\frac{\sin(nx)}{\sin(x)}\mathrm dx$$
So: $ n \geq 2$ $$ \int_{0}^{\pi}\frac{\sin((n+1)x)}{\sin(x)}\mathrm dx= \ln(\sin(x))\sin(nx)\vert_0^{\pi}-n\int_{0}^{\pi} \ln(\sin(x))\cos(nx)\mathrm dx $$
$$ =-n\int_{0}^{\pi} \ln(\sin(x))\cos(nx)\mathrm dx$$
...