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How can I compute $$\int_{-\pi}^\pi\frac{\sin(13x)}{\sin x}\cdot\frac1{1+2^x}\mathrm dx?$$

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1 Answer 1

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: $$ \int\limits_{-\pi}^{\pi} e^{ikx}dx= \begin{cases} 2\pi&\text{ if }\quad k=0\\ 0 &\text{ if }\quad k\in\mathbb{Z}\setminus\{0\} \end{cases} $$

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+1. (Write $2\pi$ in Hint 3 instead of $\pi$.) –  Did Jul 8 '12 at 22:22
    
@did Thanks!${}{}$ –  userNaN Jul 8 '12 at 22:26
    
Sorry for the complexity in my previous comment. The sum in Hint 2 should be $$\sum_{k=0}^{n-1}e^{i(n-2k-1)x}$$ –  robjohn Jul 8 '12 at 23:10
    
I was going to suggest the change you just made to Hint 1 :-) –  robjohn Jul 8 '12 at 23:15
    
@robjohn, thanks for your attention. You are always on the alert :) –  userNaN Jul 8 '12 at 23:17

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