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JEE MainMathematicsDefinite Integration

Let f: R R be a continuous function satisfying f(x) = x^2 + ₀^x e^ t-x f(t) dt for all x . Then the value of f(3) is equal to

Options

  1. A2e^3 - 8
  2. B4e^3 - 25
  3. C18
  4. D8

Correct answer

C. 18

Step-by-step solution

Given equation is f(x) = x^2 + ₀^x e^ t-x f(t) dt We can rewrite the integral by factoring out e^ -x : f(x) = x^2 + e^ -x ₀^x e^t f(t) dt Multiplying both sides by e^x to eliminate the variable x from the integral's coefficient: e^x f(x) = x^2 e^x + ₀^x e^t f(t) dt Differentiating both sides with respect to x using the product rule and Newton-Leibniz formula: e^x f(x) + e^x f'(x) = 2x e^x + x^2 e^x + e^x f(x) Canceling e^x f(x) from both sides: e^x f'(x) = (x^2 + 2x) e^x Dividing by e^x (since e^x 0 ): f'(x) = x^2

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