<p>Evaluate \(\displaystyle\int_0^{\pi/2}\log(\tan x)\,dx\)</p>
Step-by-Step Solution
Key Concept: King's rule: replace x \to \pi/2 - x, tan x \to cot x = 1/tan x, log(cot x) = -log(tan x). Adding gives 2I = 0.
<div class='solution'>
<p>Let $I = \displaystyle\int_0^{\pi/2}\log(\tan x)\,dx$.</p>
<p>By King's rule $(x\to\pi/2-x)$:</p>
<p>$$I = \int_0^{\pi/2}\log(\tan(\pi/2-x))\,dx = \int_0^{\pi/2}\log(\cot x)\,dx = \int_0^{\pi/2}\log\frac{1}{\tan x}\,dx = -\int_0^{\pi/2}\log(\tan x)\,dx = -I$$</p>
<p>$$2I = 0 \Rightarrow \boxed{I = 0}$$</p>
Correct Answer: D