Question

The integral \[ \int_0^{\frac{\pi}{2}} \log \left( \frac{\sqrt{1 - \cos 2x}}{\sqrt{1 + \cos 2x}} \right) dx \] is

• A. 1
• B. \( \frac{\pi}{4} \)
• C. 0
• D. \( \frac{\pi}{8} \)

Hint:

For integrals involving trigonometric identities, simplify the integrand using known identities and symmetry, which can often help evaluate the integral.

Answer 1

The Correct Option is C

Step 1: Simplifying the integrand.
We are asked to evaluate the integral \[ \int_0^{\frac{\pi}{2}} \log \left( \frac{\sqrt{1 - \cos 2x}}{\sqrt{1 + \cos 2x}} \right) dx. \] First, use the identity \( \cos 2x = 2 \cos^2 x - 1 \) to simplify the expression inside the logarithm.

Step 2: Using symmetry.
The integral has symmetry about \( \frac{\pi}{4} \), which allows us to conclude that the value of the integral is 0.

Step 3: Conclusion.
Thus, the value of the integral is 0, which makes option (C) the correct answer.