Question

Evaluate $\int \frac{{\tan }^7\sqrt{x}{\sec }^2\sqrt{x}}{\sqrt{x}}dx$

Answer 1

${\int }^{}\frac{{\tan }^7\sqrt{x}{\sec }^2\sqrt{x}}{\sqrt{x}}dx$

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

$\Rightarrow {\sec }^2\sqrt{x}\times \frac{1}{2\sqrt{x}}dx=dt$

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

$=2{\int }^{}{t}^{7}dt$

$=2\times \frac{t^8}{8}+c$

$=\frac{1}{4}{\tan }^8\sqrt{x}+c$