Question

The average value of f(x)=$\frac{{\cos }^{2}x}{{\sin }^{2}x+4{\cos }^{2}x}on[0,\pi /2]$ is

• A. $\pi /6$
• B. $4/\pi$
• C. $6/\pi$
• D. 1/6

Answer 1

Answer: (D)

1/6

$\frac{1}{\pi /2-0}{\int }_0^{\pi /2}\frac{{\cos }^{2}x}{{\sin }^{2}x+4{\cos }^{2}x}dx$
Substituting $t=\tan x$
$=\frac{2}{\pi }{\int }_0^{\infty }\frac{dt}{(t^2+4)(t^2+1)}=\frac{2}{\pi }\frac{1}{3}{\int }_0^{\infty }(\frac{1}{t^2+1}-\frac{1}{t^2+4})dt=\frac{1}{6}$