Definite Integration
Differentiation under integral sign
Grade 12
Question:
<p>If for a continuous function \(f(x)\), \(\int_{-\pi}^{t} (f(x) + x) \, dx = \pi^2 - t^2\), for all \(t \geq -\pi\), then \(f\!\left(-\dfrac{\pi}{3}\right)\) is equal to</p>
<p>\(\pi\)</p>
<p>\(\dfrac{\pi}{2}\)</p>
<p>\(\dfrac{\pi}{3}\)</p>
<p>\(\dfrac{\pi}{6}\)</p>
Step-by-Step Solution
Key Concept: Differentiate both sides of the integral equation with respect to t using Leibniz rule to extract f(t), then substitute the specific value.
<p><strong>Step 1:</strong> We have the equation: ∫₍₋π₎^t [f(x) + x] dx = π² - t² for all t ≥ -π</p><p><strong>Step 2:</strong> Differentiate both sides with respect to t using Leibniz rule:</p><p>d/dt[∫₍₋π₎^t (f(x) + x) dx] = d/dt[π² - t²]</p><p>This gives: f(t) + t = -2t</p><p><strong>Step 3:</strong> Solve for f(t):</p><p>f(t) = -2t - t = -3t</p><p><strong>Step 4:</strong> Substitute t = -π/3:</p><p>f(-π/3) = -3 × (-π/3) = π</p><p>∴ Answer: f(-π/3) = π</p>
Correct Answer: A