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