Question

The value of the definite integral $\underset{0}{\overset{\pi /3}{\int }}\ln (1+\sqrt{3}\tan x)dx$ is equal to

• A. $\frac{\pi }{3}\ln 2$
• B. $\frac{\pi }{3}$
• C. $\frac{{\pi }^2}{3}\ln 2$
• D. $\frac{\pi }{2}\ln 2$

Answer 1

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

$\underset{0}{\overset{\pi /3}{\int }}\ln \left(\frac{\cos x+\sqrt{3}\sin x}{\cos x}\right)dx$

$=\underset{0}{\overset{\pi /3}{\int }}\underset{\text{use king property}}{\underset{}{\ln \left(2\cos (x-\frac{\pi }{3})\right)}}dx-\underset{0}{\overset{\pi /3}{\int }}\left(\ln \cos x\right)dx$
$=\underset{0}{\overset{\pi /3}{\int }}\ln \left(2\cos x\right)dx-\underset{0}{\overset{\pi /3}{\int }}\ln \left(\cos x\right)dx$
$=\underset{0}{\overset{\pi /3}{\int }}\ln 2dx$
$=\frac{\pi }{3}\ln 2$