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Đặt $I=\int\limits_0^{\pi/2}\frac{\cos^nx}{\cos^n x+\sin^n x}dx,J=\int\limits_0^{\pi/2}\frac{\sin^n x}{\sin^n x+\cos^n x}dx$
Đặt: $x=\frac{\pi}{2}-t\Rightarrow dx=-dt$
Ta có:
$I=-\int\limits_{\pi/2}^0\frac{\cos^n(\displaystyle\frac{\pi}{2}-t)}{\sin^n(\displaystyle\frac{\pi}{2}-t)+\cos^n(\displaystyle\frac{\pi}{2}-t)}dt$
$=\int\limits_0^{\pi/2}\frac{\sin^nt}{\sin^n t+\cos^n t}dt=J$
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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