Definite Integration
Differentiation under integral sign
Grade 12
Question:
<p>If \(\int_{-\pi}^{t}(f(x)+x)\,dx = \pi^2 - t^2\), then \(f\!\left(\dfrac{\pi}{6}\right)\) equals:</p>
<p>\(\dfrac{\pi}{3}\)</p>
<p>\(\dfrac{\pi}{6}\)</p>
<p>\(0\)</p>
<p>\(-\dfrac{\pi}{3}\)</p>
Step-by-Step Solution
Key Concept: Differentiate both sides of the integral equation with respect to t using the Leibniz rule to extract f(t), then substitute t = π/6.
<p><strong>Step 1:</strong> Differentiate both sides with respect to t using Leibniz rule.</p><p>Given: ∫₍₋π₎^t [f(x) + x]dx = π² - t²</p><p>Differentiating both sides with respect to t:</p><p>f(t) + t = d/dt(π² - t²) = -2t</p><p><strong>Step 2:</strong> Solve for f(t).</p><p>f(t) + t = -2t</p><p>f(t) = -3t</p><p><strong>Step 3:</strong> Substitute t = π/6.</p><p>f(π/6) = -3 · (π/6) = -π/2</p><p>∴ Answer: A</p>
Correct Answer: A