TITLE: How to integrate $\int_0^{\pi/2} \ \frac{\cos{x}}{\sqrt{1+\cos{x}}} \, \mathrm{d}x.$
QUESTION [5 upvotes]: I need to somehow evaluate the following:
$$ \int_0^{\pi/2} \ \dfrac{\cos{x}}{\sqrt{1+\cos{x}}} \, \mathrm{d}x. $$
Can anyone give me any hints/pointers? I've tried to use parts, and some feeble substitutions, but to no avail :( 
Thanks

REPLY [5 votes]: Write $\cos x = 2\cos^2\left(\frac{x}2\right) -1$. Then $\sqrt{1+\cos x} = \sqrt 2 \cos\left(\frac x 2\right)$ when $x\in[0,\pi/2]$.
So we are seeking $$\frac{1}{\sqrt 2}\int_0^{\frac \pi 2}\frac{2\cos^2\left(\frac{x}2\right) -1}{\cos\left(\frac x 2\right)}\,dx$$
Letting $u=x/2$ then $dx=2du$ and we get the problem:
$$\sqrt 2\int_0^{\frac\pi 4} (2\cos u-\sec u)\,du$$
Integral of $\sec$ is described here.