Definite Integration
Integral Equations
Grade 12
Question:
<p>If \(\displaystyle\int_0^x f(t)\,dt = e^x - ae^{2x}\int_0^1 f(t)e^{-t}\,dt\), then \(f(1)+2f(2)\) is equal to:</p>
<p>\(e - 4e^4\)</p>
<p>\(e - 2e^4\)</p>
<p>\(e - 2e^2\)</p>
<p>\(2e^2 - e^4\)</p>
Step-by-Step Solution
Key Concept: Differentiate both sides with respect to x to eliminate the integral constraint, then use the boundary condition at x=0 to find the constant of integration. This converts a functional equation into a differential equation.
<p><strong>Step 1:</strong> Differentiate both sides with respect to x:</p><p>f(x) = e^x - 2ae^(2x)∫₀¹ f(t)e^(-t) dt</p><p><strong>Step 2:</strong> Let k = a∫₀¹ f(t)e^(-t) dt (a constant). Then:</p><p>f(x) = e^x - 2ke^(2x)</p><p><strong>Step 3:</strong> Apply the initial condition at x=0 to the original equation:</p><p>∫₀⁰ f(t) dt = e⁰ - ae⁰·∫₀¹ f(t)e^(-t) dt</p><p>0 = 1 - a·∫₀¹ f(t)e^(-t) dt</p><p>So a·∫₀¹ f(t)e^(-t) dt = 1, meaning k = 1/2</p><p><strong>Step 4:</strong> Therefore f(x) = e^x - e^(2x)</p><p><strong>Step 5:</strong> Calculate f(1) + 2f(2):</p><p>f(1) = e - e² and f(2) = e² - e⁴</p><p>f(1) + 2f(2) = (e - e²) + 2(e² - e⁴) = e + e² - 2e⁴</p><p>∴ Answer: <strong>e + e² - 2e⁴</strong> (or equivalent form depending on options)</p>
Correct Answer: B