|
|
$I=\int\limits_{0}^{\frac{\pi}{2} }\frac{\cos x}{\sqrt{7+\cos^2x} }dx +\int\limits_{0}^{\frac{\pi}{2} }\frac{1}{\cos x+2} dx $
$=\int\limits_{0}^{\frac{\pi}{2} }\frac{1}{\frac{2\sin^2 x}{15+\cos 2x} +1}d\left (\frac{\sqrt 2\sin x}{\sqrt{15+\cos 2x}}\right )+\frac{2}{\sqrt 3}\int\limits_{0}^{\frac{\pi}{2} }\frac{1}{\frac{\tan^2 x/2}{3}+1 }d\left (\frac{\tan x/2}{\sqrt 3} \right )$
$=\left[ {\arctan \left (\frac{\tan^2 x/2}{3} \right )+\frac{2}{\sqrt 3}\arctan\left (\frac{\tan x/2}{\sqrt 3} \right )} \right]_{0}^{\frac{\pi}{2} }$
$=\frac{\pi}{3 \sqrt 3}+\arctan\frac{1}{\sqrt 7}$
|