Value of $\int\limits_{\frac{\pi}{6}}^{\frac{\pi}{3}}\log(\tan x)dx$ is

0

The correct answer is Option (3) → 0

Evaluate the definite integral:

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \log(\tan x)\, dx$

Use the identity:

$\int_a^{b} \log(\tan x)\, dx = 0$ if $a + b = \frac{\pi}{2}$

Here, $\frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{2}$

So, the value of the integral is:

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \log(\tan x)\, dx = 0$