Lời giải

Đặt \( t = -x \)

\( I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{\sin^6 x + \cos^6 x}{6^x + 1} \, dx = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{6^t(\sin^6 t + \cos^6 t)}{6^t + 1} \, dt \)

\(\Rightarrow 2I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} (\sin^6 t + \cos^6 t) \, dt \quad \Rightarrow I = \frac{5\pi}{32}\)