Question

Integrate the following function with respect to $x$
$\frac{x.{\sec }^2{x}^2}{\sqrt{{\tan }^3\left(x^2\right)}}$

Answer 1

Let $I=\int \frac{x.{\sec }^2{x}^2}{\sqrt{{\tan }^3\left(x^2\right)}}.dx$
Put $\tan \left(x^2\right)=t$
$\therefore {\sec }^2\left(x^2\right)\times 2xdx=dt$
$\therefore x.se{c}^2\left(x^2\right)dx=\frac{dt}{2}$
$\therefore I=\int \frac{1}{\sqrt{t^3}}.\frac{dt}{2}$
$=\frac{1}{2}\int t^{-\frac{3}{2}}dt$
$=\frac{1}{2}$$\frac{t^{-\frac{1}{2}}}{-\frac{1}{2}}+c$
$\frac{-1}{\sqrt{t}}+c$
$\frac{-1}{\sqrt{\tan \left(x^2\right)}}+c$