# Determine the anti-derivative of $\frac{\sin(nx)}{\sin(x)}$, where $n$ is an even integer

$$\int\frac{\sin(nx)}{\sin(x)}\,dx$$

There aren't any bounds, $n$ is an even integer. I have no idea where to begin.

Hint: $\sin(z) = \frac{e^{iz}-e^{-iz}}{2i}$, hence:
$$\frac{\sin(2kx)}{\sin x} = \frac{e^{2kix}-e^{-2kix}}{e^{ix}-e^{-ix}}=\sum_{j=0}^{2k-1}e^{jix}\cdot e^{-(2k-1-j)ix}$$ and the last sum is easy to integrate. As an alternative, you may prove that: $$\frac{\sin(2kx)}{\sin(x)} = 2\cos(x)+2\cos(3x)+\ldots + 2\cos((2k-1)x)$$ since the RHS, multiplied by $\sin(x)$, becomes a telescopic sum.
• @DandyDon: how do you usually integrate $\cos(n x)$? – Jack D'Aurizio Nov 2 '15 at 12:48
• @DandyDon: it is written above. $2\cos(3x)\sin(x) = \sin(4x)-\sin(2x)$, for instance. – Jack D'Aurizio Nov 2 '15 at 17:21