Question

The value of the integral $\int_{0}^{\frac{\pi}{2}} \log(\tan \theta)\, d\theta$ is

• A. $0$
• B. $1$
• C. $\frac{\pi}{2}$
• D. $\log 2$
• E. $2$

Hint:

Use symmetry tricks for definite integrals over $[0,\frac{\pi}{2}]$.

Answer 1

The Correct Option is A

Step 1: Let the integral be \[ I = \int_{0}^{\frac{\pi}{2}} \log(\tan \theta)\, d\theta \]

Step 2: Use substitution Let: \[ \theta = \frac{\pi}{2} - x \] Then: \[ \tan\theta = \cot x \] Thus: \[ I = \int_{0}^{\frac{\pi}{2}} \log(\cot x)\, dx \]

Step 3: Add both expressions \[ 2I = \int_{0}^{\frac{\pi}{2}} \left[\log(\tan x) + \log(\cot x)\right] dx \] \[ = \int_{0}^{\frac{\pi}{2}} \log(1)\, dx \] \[ = \int_{0}^{\frac{\pi}{2}} 0\, dx = 0 \] \[ 2I = 0 \Rightarrow I = 0 \] \[ \boxed{0} \]