Question

Integrate the following function w.r.t $x$.
$\frac{{\tan }^4\sqrt{x}{\sec }^2\sqrt{x}}{\sqrt{x}}$

Answer 1

$\int \frac{{\tan }^4\sqrt{x}\cdot {\sec }^2\sqrt{x}dx}{\sqrt{x}}$

Let $\tan \sqrt{x}=t\Rightarrow \frac{{\sec }^2\sqrt{x}}{2\sqrt{x}}dx=dt\Rightarrow \frac{{\sec }^2\sqrt{x}}{\sqrt{x}}dx=2dt$

Therefore,

$\int 2t^{4}dt=\frac{2t^5}{5}+c$

$\because t=\tan \sqrt{x}$

$\therefore \int \frac{{\tan }^4\sqrt{x}\cdot {\sec }^2\sqrt{x}dx}{\sqrt{x}}=\frac{2{\tan }^5\sqrt{x}}{5}+c$