Question

If $I_1={\int }_0^{\frac{\pi }{2}}\frac{{\cos }^{2}x}{1+{\cos }^{2}x}dx,$

$I_2={\int }_0^{\frac{\pi }{2}}\frac{{\sin }^{2}x}{1+{\sin }^{2}x}dx,$

$I_3={\int }_0^{\frac{\pi }{2}}\frac{1+2{\cos }^{2}x{\sin }^{2}x}{4+2{\cos }^{2}x{\sin }^{2}x}dx$, then

• A. $I_1=I_2>I_3$
• B. $I_3>I_1=I_2$
• C. $I_1=I_2=I_3$
• D. None of these

Answer 1

The correct option is C $I_1=I_2=I_3$

$I_1={\int }_0^{\frac{\pi }{2}}\frac{{\cos }^{2}x}{1+{\cos }^{2}x}dx$
$={\int }_0^{\frac{\pi }{2}}\frac{{\cos }^2\frac{\pi }{2}-x}{1+{\cos }^2\frac{\pi }{2}-x}dx$
$={\int }_0^{\frac{\pi }{2}}\frac{{\sin }^{2}x}{1+{\sin }^{2}x}dx=I_2$
Also $I_1+I_2=={\int }_0^{\frac{\pi }{2}}(\frac{{\sin }^{2}x}{1+{\sin }^{2}x}+\frac{{\cos }^{2}x}{1+{\cos }^{2}x})dx$
$={\int }_0^{\frac{\pi }{2}}\frac{{\sin }^{2}x+{\sin }^{2}x{\cos }^{2}x+{\cos }^{2}x+{\sin }^{2}x{\cos }^{2}x}{1+{\sin }^{2}x+{\cos }^{2}x+{\sin }^{2}x{\cos }^{2}x}dx$
$=\int \frac{1+2{\sin }^{2}x{\cos }^{2}x}{2+{\sin }^{2}x{\cos }^{2}x}dx=2I_3$
$2I_1=2I_3$
$I_1=I_3$
$I_1=I_2=I_3$