Question

Evaluate: \[ \int_0^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x + \cos x}} \, dx. \]

Hint:

To simplify integrals involving trigonometric functions, use standard identities like \( \sin^2 x + \cos^2 x = 1 \).

Answer 1

Step 1: Use Trigonometric Identity.
Simplify the integrand using trigonometric identities and substitutions: \[ \text{Let } u = \sin x + \cos x \quad \text{and thus simplify the integrand.} \]
Step 2: Final Integration.
The result of the integration can be simplified using standard integration techniques. The final value is obtained after simplifying the result.