<p>The value of \(\displaystyle\int_0^{\pi/2} \log \tan x\,dx\) is ______.</p>
Step-by-Step Solution
Key Concept: Use the property that ∫₀^(π/2) f(x)dx = ∫₀^(π/2) f(π/2 - x)dx, combined with the complementary angle identity tan(π/2 - x) = cot(x), to show the integral equals its own negative.
<p><strong>Step 1:</strong> Let I = ∫₀^(π/2) log(tan x) dx</p><p><strong>Step 2:</strong> Apply the property: substitute x → (π/2 - x)</p><p>I = ∫₀^(π/2) log[tan(π/2 - x)] dx = ∫₀^(π/2) log(cot x) dx</p><p><strong>Step 3:</strong> Use cot x = 1/tan x, so log(cot x) = -log(tan x)</p><p>I = ∫₀^(π/2) [-log(tan x)] dx = -∫₀^(π/2) log(tan x) dx = -I</p><p><strong>Step 4:</strong> This means I = -I, which implies 2I = 0</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: 0