TITLE: How find the value of this integral
QUESTION [6 upvotes]: How can I compute $$\int_{-\pi}^\pi\frac{\sin(13x)}{\sin x}\cdot\frac1{1+2^x}\mathrm dx?$$

REPLY [8 votes]: Hint 1:
$$
\int\limits_{-a}^a f(x)dx=\int\limits_{-a}^a \frac{f(x)+f(-x)}{2}dx
$$
Hint 2:
$$
\frac{\sin (n x)}{\sin x}=
\frac{(e^{ix})^n-(e^{-ix})^{n}}{e^{ix}-e^{-ix}}=
\sum\limits_{k=0}^{n-1} (e^{ix})^{n-k-1}(e^{-ix})^k
$$
Hint 3:
$$
\int\limits_{-\pi}^{\pi} e^{ikx}dx=
\begin{cases}
2\pi&\text{ if }\quad k=0\\
0  &\text{ if }\quad k\in\mathbb{Z}\setminus\{0\}
\end{cases}
$$