Lời giải

Đặt \( t = \frac{\pi}{2} - x  \Rightarrow x = \frac{\pi}{2} - t \Rightarrow dx = -dt \)

\( I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^5 x}{\sin^5 x + \cos^5 x} \, dx = \int_{\frac{\pi}{2}}^{0} \frac{\sin^5 \left( \frac{\pi}{2} - t \right)}{\sin^5 \left( \frac{\pi}{2} - t \right) + \cos^5 \left( \frac{\pi}{2} - t \right)} (-dt)\)

 \( = \int_{0}^{\frac{\pi}{2}} \frac{\cos^5 t}{\cos^5 t + \sin^5 t} \, dt = J \)

•   \( I + J = \int_{0}^{\frac{\pi}{2}} \, dx = \frac{\pi}{2} \Rightarrow I = \frac{\pi}{4} \)

\( \Rightarrow \begin{cases}
n = 4\\
m = 0
\end{cases}
\Rightarrow  n + m = 4 \)

\(\Rightarrow\) Vậy chọn đáp án \(\boxed{\text{B}} \)

\( I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^n x}{\sin^n x + \cos^n x} \, dx =  \int_{0}^{\frac{\pi}{2}} \frac{\cos^n x}{\cos^n x + \sin^n x} \, dt = \frac{\pi}{4} \)