Question

The average value of $f\left(x\right)=\frac{{\cos }^{2}x}{{\sin }^{2}x+4{\cos }^{2}x}$ 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 $t=\tan x$

$=\frac{2}{\pi }{\int }_0^{\infty }\frac{dt}{(t^2+4)(t^2+1)}=\frac{2}{\pi }\frac{1}{3}{\int }_0^{\infty }(\frac{1}{t^2+1}-\frac{1}{t^2+4})dt=\frac{1}{6}$