My friend showed me this integral (and a neat way he evaluated it) and I am interested in seeing a few ways of evaluating it, especially if they are "often" used tricks. 

I can't quite recall his way, but it had something to do with an identity for phase shifting sine or cosine, like noting that $\cos(x+\pi/2)=-\sin(x)$ we get:
$$ I=\int\_{0}^{\pi/2}\frac{\cos(x)}{\sin(x)+\cos(x)}dx=\int\_{\pi/2}^{\pi}\frac{-\sin(x)}{-\sin(x)+\cos(x)}dx\\
$$
Except for as I have tried, my signs don't work out well. The end result was finding
$$ 2I=\int\_{0}^{\pi/2}dx\Rightarrow I=\pi/4
$$
Any help is appreciated! Thanks.