Tính $I = \int _{0}^{\frac{\pi }{2}}\frac{\sqrt{\cos x\left(2\cos x + 3\sin 2x \right)}}{5 - \sin x}dx$

Tính  $I =\displaystyle \int \limits_{0}^{\frac{\pi }{2}}\dfrac{\sqrt{\cos x\left(2\cos x + 3\sin 2x \right)}}{5 - \sin x}dx$

Hướng dẫn
 $I =\displaystyle \int \limits_{0}^{\frac{\pi }{2}}\dfrac{\sqrt{\cos^2 x\left(2 + 6\sin x \right)}}{5 - \sin x}dx$
$\ = \displaystyle \int \limits_{0}^{\frac{\pi }{2}}\dfrac{\cos x\sqrt{\left(2 + 6\sin x \right)}}{5 - \sin x}dx$ (do $\cos x >0$ trên $(0;\frac{\pi }{2})$)
Đặt $t= \sqrt{2 + 6\sin x}$