Question

Prove that $\int \tan x{\sec }^{2}x\sqrt{1-{\tan }^{2}x}dx=\frac{1}{3}{(1-{\tan }^{2}x)}^{3/2}.$

Answer 1

Let $I=\int \tan x{\sec }^{2}x\sqrt{1-{\tan }^{2}x}dx$

Put $1-{\tan }^{2}x=t$

$\Rightarrow -2\tan x{\sec }^{2}xdx=dt$

Therefore
$I=-\frac{1}{2}\int t^{1/2}dt$

$=-\frac{1}{2}\times \frac{2}{3}.t^{3/2}+C$

$=-\frac{1}{3}{(1-{\tan }^{2}x)}^{3/2}+C$