Question

The value of ${\int }_0^{\pi }\frac{dx}{1+2si{n}^{2}x}$ is

• A. $0$
• B. $\frac{\pi }{2\sqrt{3}}$
• C. $\frac{\pi }{\sqrt{3}}$
• D. None of these

Answer 1

Answer: (C)

$\frac{\pi }{\sqrt{3}}$

$I={\int }_0^{\pi }\frac{dx}{1+2si{n}^{2}x}$

$=2{\int }_0^{\pi /2}\frac{dx}{1+2si{n}^{2}x}$ $\because {\int }_0^{2a}f\left(x\right)dx=2{\int }_0^{a}f\left(x\right)dx$ if $f(2a-x)=f\left(x\right)$

$=2{\int }_0^{\pi /2}\frac{{\sec }^{2}xdx}{3{\tan }^{2}x+1}$

Let $t=\tan x$

$\Rightarrow {\sec }^{2}xdx=dt$

$I=2{\int }_0^{\infty }\frac{dt}{3t^2+1}$

$=\frac{2}{\sqrt{3}}ta{n}^{-1}\sqrt{3t}{\mid }_0^{\infty }$

$=\frac{\pi }{\sqrt{3}}$