Question

The value of the definite integral

\[ \int_0^{\frac{\pi}{2}} \log(\tan x) \, dx \]

is:

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

Hint:

Use symmetry properties of integrals and logarithmic identities to simplify integrals involving trigonometric functions.

Answer 1

The Correct Option is A

Step 1: To evaluate the integral, we use the property of logarithms and the symmetry of the integral: \[ I = \int_0^{\frac{\pi}{2}} \log(\tan x) \, dx. \] We can use the fact that \( \tan(\frac{\pi}{2} - x) = \cot(x) \), so: \[ I = \int_0^{\frac{\pi}{2}} \log(\cot x) \, dx. \] Step 2: Now, add the two integrals: \[ I + I = \int_0^{\frac{\pi}{2}} \log(\tan x) \, dx + \int_0^{\frac{\pi}{2}} \log(\cot x) \, dx. \] Using the identity \( \log(\tan x) + \log(\cot x) = \log(1) = 0 \), we get: \[ 2I = 0 \quad \Rightarrow \quad I = 0. \]