Question

The value of $\underset{0}{\overset{\frac{\pi }{4}}{\int }}{\tan }^6(x-[x\left]\right)+{\tan }^4(x-[x\left]\right)dx$ (where $[.]$ represents the greatest integer function)

Answer 1

For $0<x<\frac{\pi }{4}$
$\left[x\right]=0$
$\Rightarrow {\int }_0^{\frac{\pi }{4}}({\tan }^{6}x+{\tan }^{4}x)dx={\int }_0^{\frac{\pi }{4}}{\tan }^{4}x{\sec }^{2}xdx\Rightarrow I={\left(\frac{{\tan }^{5}x}{5}\right)}_0^{\frac{\pi }{4}}=0.20$