Definite Integration
King's property of definite integrals
Grade 12
Question:
<p>We have \[I = \int_0^\pi x f(\sin x)\,dx\] Using the property of definite integrals, find the value of \(I\).</p>
<p>\(\dfrac{\pi}{2}\displaystyle\int_0^\pi f(\sin x)\,dx\)</p>
<p>\(\pi\displaystyle\int_0^\pi f(\sin x)\,dx\)</p>
<p>\(\dfrac{\pi}{4}\displaystyle\int_0^\pi f(\sin x)\,dx\)</p>
<p>\(0\)</p>
Step-by-Step Solution
Key Concept: Use the substitution property: ∫₀^π f(x)dx = ∫₀^π f(π-x)dx. This transforms I into a form that allows you to add it to itself and solve for I algebraically.
<p><strong>Step 1:</strong> Apply the property ∫₀^π f(x)dx = ∫₀^π f(π-x)dx to rewrite I.</p><p>Let x → π-x in the integral:</p><p>I = ∫₀^π (π-x)f(sin(π-x))dx</p><p><strong>Step 2:</strong> Use sin(π-x) = sin(x) to simplify:</p><p>I = ∫₀^π (π-x)f(sin x)dx</p><p><strong>Step 3:</strong> Add the original integral to this transformed version:</p><p>I + I = ∫₀^π x f(sin x)dx + ∫₀^π (π-x)f(sin x)dx</p><p>2I = ∫₀^π [x + π - x]f(sin x)dx = ∫₀^π π f(sin x)dx</p><p><strong>Step 4:</strong> Solve for I:</p><p>I = (π/2)∫₀^π f(sin x)dx</p><p>∴ <strong>Answer: I = (π/2)∫₀^π f(sin x)dx</strong></p>
Correct Answer: A