TITLE: Computing $ \int_{0}^{2\pi}\frac{\sin(nx)}{\sin(x)} \mathrm dx $
QUESTION [7 upvotes]: 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$ $$ \int_{0}^{\pi}\frac{\sin((n+1)x)}{\sin(x)}\mathrm dx= \ln(\sin(x))\sin(nx)\vert_0^{\pi}-n\int_{0}^{\pi} \ln(\sin(x))\cos(nx)\mathrm dx
$$
$$ =-n\int_{0}^{\pi} \ln(\sin(x))\cos(nx)\mathrm dx$$
...

REPLY [7 votes]: $$\sin(nx) =  \sin x \cos(n-1)x + \cos x \sin(n-1)x$$
$\\$
  $$\int\frac{\sin x \cos((n-1)x) + \cos x \sin((n-1)x)}{\sin x} \text{d}x $$
or:
$$ \int (\cos((n-1)x) + \frac{\cos x \sin((n-1)x)}{\sin x})\text{d}x $$
Which equals to:
$$\frac{\sin((n-1)x)}{n-1}+\int\frac{\cos x \sin((n-1)x)}{\sin x}\text{d}x$$
So, the problem is evaluating, 
$$\int\frac{\cos x \sin((n-1)x)}{\sin x})\text{d}x$$
From some trig identities, we will get that,
$$\cos x \sin((n-1)x) = \frac{1}{2}(\sin(nx) + \sin((n-2)x)$$
Hence our "problematic" integral becomes to:
$$ \int\frac{\frac{1}{2}(\sin(nx) + \sin((n-2)x)}{\sin x} $$
Or:
$$ \frac{1}{2}\int\frac{\sin(nx)}{\sin x}+\frac{1}{2}\int\frac{\sin(n-2)x}{\sin x}$$
And if we go back and say note our original integral as:
$$ I_n=\int\frac{\sin(nx)}{\sin x}\text{d}x $$
by our above observation, we can conclude:
$$ I_n=\frac{\sin((n-1)x}{n-1}+\frac{1}{2}I_n+\frac{1}{2}I_{n-2}$$
or:
$$ I_n=2 \frac{\sin((n-1)x)}{n-1}+I_{n-2}$$
Could you proceed?