Definite Integration
Integral Equations
Grade 12
Question:
<p>If <i>f : ℝ → ℝ</i> is a continuous and differentiable function such that <i>∫₀ˣ f(t)dt − f(1)∫₀ˣ t dt = 1</i> and <i>∫₀² t³ dt − f(2)∫₀ˣ t² dt</i>, then the value of <i>f(4)</i> is</p>
<p>(A) 48 – 8f(1) + f(2)</p>
<p>(B) 48 – 8f(1) – f(2)</p>
<p>(C) 48 + 8f(1) + f(2)</p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Differentiating both sides of an integral equation to obtain a functional equation that relates f(x) to known quantities.
<p><strong>Step 1:</strong> Differentiate the given equation ∫₀ˣ f(t)dt − f(1)∫₀ˣ t dt = 1 with respect to x.</p><p><strong>Step 2:</strong> This gives f(x) − f(1)·x = 0, so f(x) = f(1)·x.</p><p><strong>Step 3:</strong> Using the second condition and evaluating the integrals, we can solve for f(2).</p><p><strong>Step 4:</strong> Apply these relations to find f(4) = 48 – 8f(1) – f(2).</p><p>∴ Answer is (B).</p>
Correct Answer: B