$\underset{0}{\overset{\pi /2}{\int }}\frac{\cos x}{\sqrt{1+\cos x\sin x}}dx$

$I=\underset{0}{\overset{\pi /2}{\int }}\frac{\cos x}{\sqrt{1+\cos xsinx}}dx\to \left(1\right)$

$I=\underset{0}{\overset{\pi /2}{\int }}\frac{\sin x}{\sqrt{1+\sin x\cos x}}dx\to \left(2\right)$

$f\left(a-x\right)=f\left(x\right)$

$I+I=\underset{0}{\overset{\pi /2}{\int }}\frac{\left(\cos x+\sin x\right)}{\sqrt{1+\cos x\sin x}}dx$

$=\underset{0}{\overset{\pi /2}{\int }}\frac{\sqrt{2}\left(\cos x+\sin x\right)}{\sqrt{2+2\cos x\sin x}}dx$

$2I=\sqrt{2}\underset{0}{\overset{\pi /2}{\int }}\frac{\left(\cos x+\sin x\right)}{\sqrt{3-{\left(\sin x-\cos x\right)}^2}}dx$

$=\frac{1}{\sqrt{3}}{\sin }^{-1}\frac{\left(\sin x-\cos x\right)}{\sqrt{3}}{]}_0^{\pi /2}$