TITLE: Evaluate $ \int^{ \pi/2}_{- \pi/2} \frac {1}{ 1+e^{\sin x} }dx $
QUESTION [6 upvotes]: Evaluate $ \int^{\pi/2}_{-\pi/2} \frac {1}{ 1+e^{\sin x} }dx $
Solution:

I think odd, even functions are of no use here.
Also we get nothing by taking $e^{\sin x} $ common in denominator.
Also rationalizing  the denominator is of no use.

Really I have no idea how to solve this question.
Please help.

REPLY [2 votes]: Generalization:
Use $\displaystyle I=\int_p^qf(x)\ dx=\int_p^qf(p+q-x)\ dx$
so that $\displaystyle I+I=\int_p^q[f(x)+f(p+q-x)]dx$
For $\displaystyle I=\int_{-a}^a\dfrac1{1+c^{g(x)}}$ where $g(x)$ is any odd function i.e., $g(-x)=-g(x)$ like $\sin x$
$f(p+q-x)=\dfrac1{1+c^{g(-x)}}=\dfrac{c^{g(x)}}{1+c^{g(x)}}$
$f(x)+f(p+q-x)=1$
The rest should be easy.