Question

$\underset{0}{\overset{\pi /2}{\int }}\frac{\sin x\cos x}{1+{\sin }^{4}x}dx=$

Answer 1

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

Let ${\sin }^{2}x=t$ at $x=0\Rightarrow t=0$

Differentiating w.r.t $x$ at $x=x/2\Rightarrow t=2$

$2\sin x\cos xdx=dt$

$\sin x\cos xdx=dt/2$

$\therefore I=\frac{1}{2}{\int }_0^1\frac{dt}{({\sin }^{2}x{)}^2+1}$

$=\frac{1}{2}{\int }_0^1\frac{dt}{t^2+1}$

$=\frac{1}{2}[{\tan }^{-1}t{]}_0^1$

$=\frac{1}{2}[{\tan }^{-1}1-{\tan }^{-1}0]=\frac{1}{2}[\frac{\pi }{4}=0]$

$=\frac{\pi }{8}$