Question

Evaluate ${\int }_0^{\pi /2}\frac{dx}{2+\sin 2x}$

• A. $\frac{2\pi }{3}$
• B. $\frac{\pi }{3}$
• C. $\frac{2\pi }{5}$
• D. None of these
  

Answer 1

The correct option is A $\frac{2\pi }{3}$

${\int }_0^{2\pi }\frac{dx}{2+\sin 2x}={\int }_0^{2\pi }\frac{dx}{2+\frac{2\tan x}{1+{\tan }^{2}x}}dx$
$=\frac{1}{2}{\int }_0^{2x}\frac{{\sec }^{2}x}{{\tan }^{2}x+\tan x+1}dx$
Put $\tan x=t\Rightarrow {\sec }^{2}xdx=dt$
$I=\frac{1}{2}\int \frac{1}{t^2+t+1}dt.=\frac{1}{2}\int \frac{1}{{(t+\frac{1}{2})}^2-\frac{3}{2}}=\frac{2\pi }{3}$