Question

\(\int_{0}^{\frac{\pi}{2}} \log (\tan x) \, dx = ? \)

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{2} \)
• C. 0
• D. \( \pi \)

Hint:

Whenever you encounter integrals with logarithmic functions involving trigonometric identities, look for symmetry to simplify the calculation.

Answer 1

The Correct Option is C

We are asked to evaluate the integral: \[ I = \int_{0}^{\frac{\pi}{2}} \log (\tan x) \, dx \] Step 1: Use the symmetry of the tangent function First, recall that \(\tan \left( \frac{\pi}{2} - x \right) = \cot x\), which implies: \[ \log (\tan \left( \frac{\pi}{2} - x \right)) = \log (\cot x) = -\log (\tan x) \] Step 2: Split the integral We can use this symmetry to split the integral: \[ I = \int_{0}^{\frac{\pi}{2}} \log (\tan x) \, dx = \int_{0}^{\frac{\pi}{2}} \log (\cot x) \, dx \] Adding these two expressions for \(I\): \[ 2I = \int_{0}^{\frac{\pi}{2}} \log (\tan x) + \log (\cot x) \, dx \] Since \(\log (\tan x) + \log (\cot x) = 0\), we have: \[ 2I = \int_{0}^{\frac{\pi}{2}} 0 \, dx = 0 \] Thus, \(I = 0\). Therefore, the correct answer is: (C) 0.