Definite Integration
King's property of definite integrals
Grade 12
Question:
<p>If <br>\(\int_0^{\pi} x f(\sin x)\,dx = A\int_0^{\pi/2} f(\sin x)\,dx\),<br> then the value of \(A\) is:</p>
<p>\(0\)</p>
<p>\(\pi\)</p>
<p>\(\dfrac{\pi}{4}\)</p>
<p>\(2\pi\)</p>
Step-by-Step Solution
Key Concept: Use the property that ∫₀^π f(sin x)dx = 2∫₀^(π/2) f(sin x)dx by substituting x = π - t to transform the left side integral and exploit symmetry of the sine function.
<p><strong>Step 1:</strong> Split the integral: ∫₀^π x f(sin x)dx = ∫₀^(π/2) x f(sin x)dx + ∫_(π/2)^π x f(sin x)dx</p><p><strong>Step 2:</strong> For the second integral, substitute x = π - t, so dx = -dt. When x = π/2, t = π/2; when x = π, t = 0.</p><p>∫_(π/2)^π x f(sin x)dx = ∫_(π/2)^0 (π - t)f(sin(π - t))(-dt) = ∫_0^(π/2) (π - t)f(sin t)dt</p><p><strong>Step 3:</strong> Since sin(π - t) = sin t, we have:</p><p>∫₀^π x f(sin x)dx = ∫₀^(π/2) x f(sin x)dx + ∫_0^(π/2) (π - x)f(sin x)dx</p><p>= ∫₀^(π/2) [x + π - x]f(sin x)dx = π∫₀^(π/2) f(sin x)dx</p><p><strong>Step 4:</strong> Comparing with A∫₀^(π/2) f(sin x)dx, we get A = π</p><p>∴ Answer: <strong>A = π</strong></p>
Correct Answer: B