Question

Integrate $\int \frac{1}{{\sin }^{4}x+{\cos }^{4}x}dx$

Answer 1

$\Rightarrow \int \frac{dt}{2+t^2}$

$\Rightarrow \frac{1}{\sqrt{2}}{\tan }^{-1}\left(\frac{t}{\sqrt{2}}\right)+constatnt$

$\Rightarrow \frac{1}{{\sin }^{4}x+{\cos }^{4}x}dx=\frac{1}{\sqrt{2}}{\tan }^{-1}\left(\frac{\tan \left(2n\right)}{\sqrt{2}}\right)+constant$

$\int \frac{1}{{\sin }^{4}x+{\cos }^{4}x}$

we know

${\sin }^{4}x+{\cos }^{4}x=1-2{\sin }^{2}x{\cos }^{2}x$

$=1-\frac{1}{2}{\sin }^{2}2x$

$=\frac{1}{4}(3+\cos 4x)$

let $t=\tan \left(2x\right)$

$\therefore \int \frac{1}{{\sin }^{4}x+{\cos }^{4}x}dx=\int \frac{4}{3+\frac{1-t^2}{1+t^2}}.\frac{dt}{2(1+t^2)}$

$=\int \frac{4(1+t^2)}{3+3t^2+1-t^2}.\frac{dt}{2(1+t^2)}$

$=\int \frac{4}{(4+2t^2)}\frac{dt}{2}$