By using the properties of definite integrals, evaluate the integrals`int_0^(pi/2)(2logsinx-logsin2x)dx`

Explanation

To evaluate the integral \( I = \int_0^{\frac{\pi}{2}} (2 \log \sin x - \log \sin 2x) \, dx \), we can follow these steps:

### Step 1: Simplify the integrand
We start by rewriting the integrand:
\[ 2 \log \sin x - \log \sin 2x \]
Using the property of logarithms, we know that \( \log a - \log b = \log \frac{a}{b} \). Also, we can express \( \sin 2x \) as \( 2 \sin x \cos x \):
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