Question

Evaluate \[ \int_0^{\frac{\pi}{2}} \log \sin x \, dx \]

Hint:

The integral \( \int_0^{\frac{\pi}{2}} \log \sin x \, dx \) is a standard result and evaluates to \( -\frac{\pi}{2} \log 2 \).

Answer 1

Step 1: Recognize the standard integral. 
The integral \( \int_0^{\frac{\pi}{2}} \log \sin x \, dx \) is a standard result and is known to evaluate to: \[ \int_0^{\frac{\pi}{2}} \log \sin x \, dx = -\frac{\pi}{2} \log 2 \] Step 2: Conclusion. 
Thus, the value of the integral is: \[ -\frac{\pi}{2} \log 2 \]