The question is:

Find the definite integral:

$\displaystyle \int\limits_0^{\pi/2} \frac{1}{1+cos(\alpha)cos(x)}dx$

($\displaystyle \alpha$ is a constant)

I've tried using the t-substitution, but I got quite a messy answer.

Possibly there's some simplification involving the fact that the constant is written in terms of cos that I'm not seeing?

The solution given isSpoiler:

Thanks.