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