Let, I n = ∫ tan n x d x n > 1 . If I 4 + I 6 = a tan 5 x + b x 5 + c , then the ordered pair a ,…

Given, ${\mathrm{ I}}_n=\int {\tan }^{\mathrm{n}}xdx$

$\Rightarrow I_4+I_6=\int {\tan }^{4}xdx+\int {\tan }^{6}xdx$

$\Rightarrow I_4+I_6=\int {\tan }^{4}xdx\left(1+{\tan }^{2}x\right)dx$

$\Rightarrow I_4+I_6=\int {\tan }^{4}x{\sec }^{2}xdx$

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

$\Rightarrow I_4+I_6=\int t^{4}dt$

$\Rightarrow I_4+I_6=\frac{t^5}{5}+c$, where $c$ is the constant of integration.

$\Rightarrow a{\tan }^{5}x+b{x}^5+c=\frac{{\tan }^{5}x}{5}+c$

$\therefore a=\frac{1}{5}\& b=0$