TỔNG QUÁT
Đặt $x=\dfrac{\pi}{2} - t\Rightarrow dx=-dt$
$I=\int_0^{\frac{\pi}{2}} \dfrac{\sqrt[n]{\cos t}}{\sqrt[n]{\sin t} +\sqrt[n]{\cos t}}dt=\int_0^{\frac{\pi}{2}} \dfrac{\sqrt[n]{\cos x}}{\sqrt[n]{\sin x} +\sqrt[n]{\cos x}}dx$
$\Rightarrow 2I = \int_0^{\frac{\pi}{2}} \dfrac{\sqrt[n]{\sin x}}{\sqrt[n]{\sin x} +\sqrt[n]{\cos x}}dx +\int_0^{\frac{\pi}{2}} \dfrac{\sqrt[n]{\cos x}}{\sqrt[n]{\sin x} +\sqrt[n]{\cos x}}dx=\int_0^{\frac{\pi}{2}}dx$
$\Rightarrow I = \dfrac{\pi}{4}$