Question

Evaluate : ${\int }_0^{\pi /4}{\tan }^{6}xdx$

Answer 1

$\begin{array}{c}{\int }_0^{\pi /4}{\tan }^{6}xdx\\ \Rightarrow {\int }_0^{\pi /2}{\tan }^{2}x\cdot {\sec }^{2}xdx-{\int }_0^{gp/2}{\tan }^{4}xdx\\ let\left(\tan x\right)=t\\ {\sec }^{2}xdx=dt\\ \Rightarrow {\int }_0^{\pi /4}{t}^{4}dt-{\int }_0^{\pi /4}{\sec }^{2}x\cdot {\tan }^{2}x-{\int }_0^{\pi /4}{\tan }^{2}x\\ \Rightarrow {\left[\frac{t^5}{5}\right]}^{\pi /4}-{\int }_0^{\pi /4}{t}^{2}dt-{\int }_0^{\pi /4}{\sec }^{2}xdx-{\int }_0^{\pi /4}dx\\ \Rightarrow \left[\frac{1}{5}-\frac{1}{3}+1-\pi /4\right]\\ \Rightarrow \left[\frac{13}{15}-\pi /4\right]\end{array}$