<p>Let \( I = \int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \). Find the value of \(I\).</p>
Step-by-Step Solution
Key Concept: Use the symmetry property: if I = ∫₀^(π/2) f(x)dx, then also I = ∫₀^(π/2) f(π/2 - x)dx. Adding these two expressions yields a constant integrand that evaluates easily.
<p><strong>Step 1:</strong> Let I = ∫₀^(π/2) √sin x / (√sin x + √cos x) dx</p><p><strong>Step 2:</strong> Use the property: substitute x → (π/2 - x). Then sin x ↔ cos x, dx remains same.</p><p>I = ∫₀^(π/2) √cos x / (√cos x + √sin x) dx</p><p><strong>Step 3:</strong> Add the two expressions for I:</p><p>2I = ∫₀^(π/2) [√sin x + √cos x] / [√sin x + √cos x] dx</p><p>2I = ∫₀^(π/2) 1 dx = π/2</p><p><strong>Step 4:</strong> Therefore, I = π/4 ≈ 0.7854</p><p>∴ Answer: <strong>0.7857</strong> (or π/4)</p>
Correct Answer: 0.7857