<p>The value of \(\displaystyle\int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}\,dx\) is ______ (up to four decimal places).</p>
Step-by-Step Solution
Key Concept: Use the property that for ∫₀^(π/2) f(x)dx, substituting x → (π/2 - x) gives ∫₀^(π/2) f(π/2 - x)dx, then add both forms to find the integral equals π/4. This works because √sin x and √cos x swap under this substitution.
<p><strong>Step 1:</strong> Let I = ∫₀^(π/2) √sin x/(√sin x + √cos x) dx</p><p><strong>Step 2:</strong> Use substitution x → π/2 - x. Then sin x → cos x and cos x → sin x, giving:</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 + √cos x]/(√sin x + √cos x) dx = ∫₀^(π/2) 1 dx = π/2</p><p><strong>Step 4:</strong> Therefore I = π/4 = 3.14159.../4</p><p>∴ Answer: <strong>0.7854</strong></p>
Correct Answer: 0.7854