How can I compute $\int_{-\pi}^\pi\frac{\sin(13x)}{\sin x}\cdot\frac1{1+2^x}\mathrm dx?$
How find the value of this integral
6
$\begingroup$
calculus
integration
1 Answers
8
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} $
-
0@robjohn Thanks! – 2012-07-08