Definite Integration
King's property
Grade 12
Question:
<p>Evaluate \(\int_0^{\pi/2} \dfrac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}\, dx\)</p>
<p>\(\dfrac{\pi}{4}\)</p>
<p>\(\dfrac{\pi}{2}\)</p>
<p>zero</p>
<p>1</p>
Step-by-Step Solution
Key Concept: Use the property that ∫₀^(π/2) f(x)dx = ∫₀^(π/2) f(π/2 - x)dx, then add the original and transformed integrals to eliminate the complex radicals.
<p><strong>Step 1:</strong> Let I = ∫₀^(π/2) √(sin x)/(√(sin x) + √(cos x)) dx</p><p><strong>Step 2:</strong> Apply the property: substitute x → (π/2 - x), so sin x ↔ cos x. This gives:</p><p>I = ∫₀^(π/2) √(cos x)/(√(cos x) + √(sin x)) dx</p><p><strong>Step 3:</strong> Add both expressions:</p><p>2I = ∫₀^(π/2) [√(sin x)/(√(sin x) + √(cos x)) + √(cos x)/(√(sin x) + √(cos x))] dx</p><p><strong>Step 4:</strong> Simplify the numerator:</p><p>2I = ∫₀^(π/2) (√(sin x) + √(cos x))/(√(sin x) + √(cos x)) dx = ∫₀^(π/2) 1 dx</p><p><strong>Step 5:</strong> Evaluate:</p><p>2I = [x]₀^(π/2) = π/2</p><p>∴ I = <strong>π/4</strong></p>
Correct Answer: A