Definite Integration
Properties of definite integrals
Grade 12
Question:
<p>Evaluate \[I = \int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}\,dx\]</p>
<p>\(\dfrac{\pi}{4}\)</p>
<p>\(\dfrac{\pi}{2}\)</p>
<p>\(\pi\)</p>
<p>0</p>
Step-by-Step Solution
Key Concept: Use the property that I + J = π/4 where J is the same integral with sin and cos swapped. Since the integrand structure is symmetric under the transformation x → π/2 - x, adding I and J yields a constant that can be easily evaluated.
<p><strong>Step 1:</strong> Let I = ∫₀^(π/2) √sin x/(√sin x + √cos x) dx</p><p><strong>Step 2:</strong> Consider the complementary integral J = ∫₀^(π/2) √cos x/(√sin x + √cos x) dx</p><p><strong>Step 3:</strong> Add I and J: I + J = ∫₀^(π/2) (√sin x + √cos x)/(√sin x + √cos x) dx = ∫₀^(π/2) 1 dx = π/2</p><p><strong>Step 4:</strong> Use the substitution x → π/2 - x in J. This gives J = ∫₀^(π/2) √sin(π/2 - x)/(√sin(π/2 - x) + √cos(π/2 - x)) dx = ∫₀^(π/2) √cos x/(√cos x + √sin x) dx, which equals I by relabeling.</p><p><strong>Step 5:</strong> Therefore I = J. From I + J = π/2, we get 2I = π/2</p><p>∴ <strong>Answer: I = π/4</strong></p>
Correct Answer: A