Đặt $t = - x$ $\Rightarrow I = \int_{\frac{\pi }{4}}^{\frac{- \pi }{4}}\frac{sin^{6}x + cos^{6}x}{1 + 6^{ - x}} dx$
$\Rightarrow I = \int_{\frac{\pi }{4}}^{\frac{ - \pi }{4}}\frac{sin^{6}x + cos^{6}x}{6^{x} + 1}.6^{x} dx$
$\Rightarrow 2I = \int_{\frac{\pi }{4}}^{\frac{ - \pi }{4}} \left(sin^{6}x + cos^{6}x\right) dx$
$\Rightarrow 2I = \int_{\frac{\pi }{4}}^{- \frac{\pi }{4}} \left( \frac{5}{8} + \frac{3}{8}cos4x\right) dx$
$\Rightarrow 2I = \frac{-5\pi }{16} \Rightarrow I = \frac{-5\pi }{32}$