Đặt: $I=\int\limits_{0}^{\frac{\pi}{4}}\dfrac{\sin^52x}{\sin^52x+\cos^52x}dx;J=\int\limits_{0}^{\frac{\pi}{4}}\dfrac{\cos^52x}{\sin^52x+\cos^52x}dx$
Đặt: $\dfrac{\pi}{4}-x=t \Rightarrow dx=dt$
Đổi cận: $x=0 \Rightarrow t=\dfrac{\pi}{4}$
$x=\dfrac{\pi}{4} \Rightarrow t=0$
Ta có:
$I=\int\limits_{0}^{\frac{\pi}{4}}\dfrac{\sin^52x}{\sin^52x+\cos^52x}dx$
$=-\int\limits_{\frac{\pi}{4}}^0 \dfrac{\sin^5(\dfrac{\pi}{2}-2t)}{\sin^5(\dfrac{\pi}{2}-2t)+\cos^5(\dfrac{\pi}{2}-2t)}dt$
$=\int\limits_{0}^{\frac{\pi}{4}}\dfrac{\cos^52t}{\cos^52t+\sin^52t}dt=J$
Mà: $I+J=\int\limits_0^{\frac{\pi}{4}}dt=\dfrac{\pi}{4} \Rightarrow I=J=\dfrac{\pi}{8}$