Question

Evaluate the definite integral ${\int }_0^{\frac{\pi }{4}}\frac{\sin x\cos x}{{\cos }^{4}x+{\sin }^{4}x}dx$

Answer 1

Let $I={\int }_0^{\frac{\pi }{4}}\frac{\sin x\cos x}{{\cos }^{4}x+{\sin }^{4}x}dx$
$\Rightarrow I={\int }_0^{\frac{\pi }{4}}\frac{\frac{\left(\sin x\cos x\right)}{{\cos }^{4}x}}{\frac{({\cos }^{4}x+{\sin }^{4}x)}{{\cos }^{4}x}}dx$
$\Rightarrow I={\int }_0^{\frac{\pi }{4}}\frac{\tan x{\sec }^{2}x}{1+{\tan }^{4}x}dx$
Let ${\tan }^{2}x=t\Rightarrow 2\tan x{\sec }^{2}xdx=dt$
When $x=0,t=0$ and when $x=\frac{\pi }{4},t=1$
$\therefore I=\frac{1}{2}{\int }_0^1\frac{dt}{1+t^2}$
$=\frac{1}{2}[{\tan }^{-1}t{]}_0^1$
$=\frac{1}{2}[{\tan }^{-1}1-{\tan }^{-1}0]$
$=\frac{1}{2}\left[\frac{\pi }{4}\right]=\frac{\pi }{8}$