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Đặt $I=\int\limits_0^{\pi/2}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx,J=\int\limits_0^{\pi/2}\frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$
Đặt: $x=\frac{\pi}{2}-t\Rightarrow dx=-dt$
Ta có:
$I=-\int\limits_{\pi/2}^0\frac{\sqrt{\sin(\displaystyle\frac{\pi}{2}-t)}}{\sqrt{\sin(\displaystyle\frac{\pi}{2}-t)}+\sqrt{\cos(\displaystyle\frac{\pi}{2}-t)}}dt$
$=\int\limits_0^{\pi/2}\frac{\sqrt{\cos t}}{\sqrt{\sin t}+\sqrt{\cos t}}dt=J$
Mà ta có:
$I+J=\int\limits_0^{\pi/2}dx=\frac{\pi}{2}$
Suy ra: $I=J=\frac{\pi}{4}$
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