Question

Find the value of $\int {\sec }^{2}x{\tan }^{3}xdx$

Answer 1

Let $I=\int {\sec }^{2}x{\tan }^{3}xdx$

Let $\tan x=t$

differentiate with respect to $x$

$\frac{d}{dx}\tan x=\frac{dt}{dx}$

${\sec }^{2}x=\frac{dt}{dx}$

$dt={\sec }^{2}xdx$

$I=\int t^{3}dt$

$I=\frac{t^4}{4}+c$

$I=\frac{{\tan }^{4}x}{4}+c$