Question

Evaluate:

${\int }_0^{\pi /2}\frac{x\sin x\cos x}{{\cos }^{4}x+{\sin }^{4}x}dx$.

Answer 1

$I={\int }_0^{\pi /2}\frac{(\pi /2-x)\sin x\cos x}{{\sin }^{4}x+{\cos }^{4}x}dx$ from properties

$2I=\frac{\pi }{4}{\int }_0^{\pi /2}\frac{2\sin x\cos x}{{\sin }^{4}x+{\cos }^{4}x}dx$

Putting ${\sin }^{2}x=m$

$2\sin x\cos xdx=dm$

$2I=\frac{\pi }{4}{\int }_0^1\frac{dm}{(1-m{)}^2+m^2}$

$I=\frac{\pi }{8}{\int }_0^1\frac{dm}{(m-\frac{1}{2}{)}^2+\frac{1}{4}}$

$I=\frac{\pi }{4}[{\tan }^{-1}(2m-1){]}_0^1$

$I=-\frac{{\pi }^2}{8}$