Let, I _ n = ∫ ^ n xdx (n > 1) . If I _ 4 + I _ 6 = a ^ 5 x + b x ^...

Given, ${I}_{n} = ∫ \tan^{n}xdx$

$⇒ {I}_{4} + {I}_{6} = ∫ \tan^{4}xdx + ∫ \tan^{6}xdx$

$⇒ {I}_{4} + {I}_{6} = ∫ \tan^{4}xdx\left(1 + \tan^{2}x\right)dx$

$⇒ {I}_{4} + {I}_{6} = ∫ \tan^{4}x{sec}^{2}xdx$

Put, $\tan x = t ⇒ \sec^{2}xdx = dt$

$⇒ {I}_{4} + {I}_{6} = ∫ {t}^{4}dt$

$⇒ {I}_{4} + {I}_{6} = \dfrac{{t}^{5}}{5} + c$, where $c$ is the constant of integration.

$⇒ a\tan^{5}x + b{x}^{5} + c = \dfrac{\tan^{5}x}{5} + c$

$∴ a = \dfrac{1}{5} \text{ and } b = 0$