Question

The value of definite integral \( \int_0^{\pi/2} \log(\tan x) dx \) is:

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

Hint:

When solving integrals of logarithmic functions, try applying symmetry and properties of the functions to simplify the process.

Answer 1

The Correct Option is A

Step 1: Define \(I = \int_0^{\pi/2} \log(\tan x) dx\). Utilize the property \(\int_a^b f(x) dx = \int_a^b f(a + b - x) dx\). Consequently, \(I = \int_0^{\pi/2} \log(\tan(\frac{\pi}{2} - x)) dx = \int_0^{\pi/2} \log(\cot x) dx\).

Step 2: Summing the two expressions for \(I\) yields \(2I = \int_0^{\pi/2} [\log(\tan x) + \log(\cot x)] dx\). This simplifies to \(2I = \int_0^{\pi/2} \log(1) dx = 0\).

Thus, \( I = 0 \).