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JEE MainMathematicsDefinite Integration

Let f be a continuous function on R satisfying f(x) = 4x + 2 + ₀^x e^ x-t f(t) dt . Then the value of f( 3) - 2 3 is equal to _______.

Correct answer

18

Step-by-step solution

Given, f(x) = 4x + 2 + ₀^x e^ x-t f(t) dt f(x) = 4x + 2 + e^x ₀^x e^ -t f(t) dt Substituting x = 0 , we get: f(0) = 2 Differentiating with respect to x using the product rule and Leibniz's rule: f'(x) = 4 + e^x ₀^x e^ -t f(t) dt + e^x e^ -x f(x) From the original equation, e^x ₀^x e^ -t f(t) dt = f(x) - 4x - 2 . Substituting this into the derivative: f'(x) = 4 + (f(x) - 4x - 2) + f(x) f'(x) = 2f(x) - 4x + 2 f'(x) - 2f(x) = -4x + 2 This is a linear differential equation of the form dy dx + Py = Q . Integrating Facto

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