Evaluate: `int_0^(pi//4)sqrt(1+sin2x)dx` (ii) `int_0^(pi//4)sqrt(1-sin2x)dx`

Explanation

To evaluate the integrals \( I_1 = \int_0^{\frac{\pi}{4}} \sqrt{1 + \sin 2x} \, dx \) and \( I_2 = \int_0^{\frac{\pi}{4}} \sqrt{1 - \sin 2x} \, dx \), we will follow these steps:

### Part 1: Evaluating \( I_1 = \int_0^{\frac{\pi}{4}} \sqrt{1 + \sin 2x} \, dx \)

1. **Rewrite the integrand**:
\[ \sqrt{1 + \sin 2x} = \sqrt{1 + 2 \sin x \cos x} = \sqrt{\sin^2 x + \cos^2 x + 2 \sin x \cos x} \]
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