Value of $\int\limits_{\frac{\pi}{6}}^{\frac{\pi}{3}}\log(\tan x)dx$ is
Answer & explanation
Correct answer: option 3
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:
$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \log(\tan x)\, dx = 0$