Question

Evaluate: \[ \int_0^{\pi/2} \log (\cos x) \, dx. \]

Hint:

Use symmetry and known definite integrals involving logarithms of trigonometric functions.

Answer 1

This is a standard integral. We know that: \[ \int_0^{\pi/2} \log(\sin x) \, dx = \int_0^{\pi/2} \log(\cos x) \, dx = -\frac{\pi}{2} \log 2. \] Hence, \[ \int_0^{\pi/2} \log (\cos x) \, dx = -\frac{\pi}{2} \log 2. \]
Final answer: \[ \boxed{ \int_0^{\pi/2} \log (\cos x) \, dx = -\frac{\pi}{2} \log 2. } \]