# I Integral of a special trigonometic functions

1. Sep 21, 2016

### KFC

Hi all,
I am working on the following integral

$\int_0^{2\pi}\frac{1+\cos[\alpha x]}{1+\sin x}dx$

where $\alpha$ is odd integer. Unless I set the $\alpha$ to a number then I can find the integral with mathematica easily. For general case with symbolic $\alpha$, I cannot find the integral like this kind from any table and mathematica won't give the result as well. I am trying to apply the following relation to simplify the integrand

$\cos[\alpha x] = 2\cos x \cos[(\alpha-1)x] - \cos[(\alpha-2)x]$

it doesn't help to compute the integral. Any idea is welcomed. Thanks.

2. Sep 22, 2016

### Orodruin

Staff Emeritus