Different ways of evaluating $\int_{0}^{\pi/2}\frac{\cos(x)}{\sin(x)+\cos(x)}dx$ 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. 

You're integrating on $[0, \pi/2]$ so replacing $x$ by $\pi/2 -x$ we see that $$I=\int_0^{\pi/2} \frac{\sin{x}}{\cos{x}+\sin{x}}dx.$$ Now sum this integral with the initial expression and notice that $2I=\pi/2$ hence...
