Question

Solve $\int \left(\frac{\tan xdx}{\sqrt{{\sin }^{4}x+{\cos }^{4}x}}\right)$

Answer 1

$\begin{array}{c}{\int }^{}\frac{\tan x}{\sqrt{{\sin }^{4}x+{\cos }^{4}x}}\\ ={\int }^{}\frac{\tan x}{{\cos }^{2}x\sqrt{{\tan }^{4}x+1}}dx\\ ={\int }^{}\frac{\tan x.{\sec }^{2}x}{\sqrt{{\tan }^{4}x+1}}\\ put\tan x=t\\ {\sec }^{2}xdx=dt\\ ={\int }^{}\frac{t.dt}{\sqrt{t^4+1}}\\ Againput{t}^2=p\\ 2tdt=dp\\ =\frac{1}{2}{\int }^{}\frac{dp}{\sqrt{p^2+1}}\\ Hence,weget\\ I=\frac{1}{2}\left\{{\tan }^{2}x+\sqrt{1+{\tan }^{4}x}\right\}+C\end{array}$

Hence, this is the answer.