Question

The area between the curves $y=\tan x,y=2\sin x$ and $x$-axis in $-\frac{\pi }{3}\le x\le \frac{\pi }{3}$ is

• A. $2-\ln 2$
• B. $3-ln2$
• C. $4-\ln 2$
• D. None of these

Answer 1

The correct option is A $2-\ln 2$

Due to symmetry

$2{\int }_0^{\frac{\pi }{3}}2\sin x-\tan xdx$

$=2[-2\cos x+\ln \cos x{]}_0^{\frac{\pi }{3}}$

$|2\ln |\cos x|1-4\cos x{|}_0^{\frac{\pi }{3}}$

$=2-2\ln 2$