Question

Obtain reduction formula : $\int {\tan }^{n}xdx$
for integer $n\ge 2$ and evaluate : $\int {\tan }^{6}xdx$

Answer 1

$\int {\tan }^{n}xdx$

$=\int {\tan }^{n-2}x\left({\tan }^{2}x\right)dx$

$=\int {\tan }^{n-2}x\left({\sec }^{2}x\right)dx-\int {\tan }^{n-2}xdx$

Let$\tan x=t,{\sec }^{2}xdx=dt$

$=\int t^{n-2}dt-\int {\tan }^{n-2}xdx$

$=\frac{t^{n-1}}{n-1}-\int {\tan }^{n-2}xdx$

So $\int {\tan }^{n}xdx$$=\frac{{\tan }^{n-1}x}{n-1}-\int {\tan }^{n-2}xdx$

So $\int {\tan }^{6}xdx$$=\frac{{\tan }^{5}x}{5}-\int {\tan }^{4}xdx$

$=\frac{{\tan }^{5}x}{5}-\frac{{\tan }^{3}x}{3}+\int {\tan }^{2}xdx$

$=\frac{{\tan }^{5}x}{5}-\frac{{\tan }^{3}x}{3}+\int ({\sec }^{2}x-1)dx$

$=\frac{{\tan }^{5}x}{5}-\frac{{\tan }^{3}x}{3}+\tan x-x+c$