Let f: (0, ) R be a differentiable function such that ₁^x t^2 f(t) dt = x f(x) - e for all x 0 . Then the value of f(2) is equal to

Step-by-step solution

Given the integral equation: ₁^x t^2 f(t) dt = x f(x) - e First, find the initial condition by substituting x = 1 into the equation: ₁^1 t^2 f(t) dt = 1 f(1) - e 0 = f(1) - e f(1) = e Next, differentiate both sides of the integral equation with respect to x using the Leibniz rule: x^2 f(x) = f(x) + x f^ (x) Rearranging the terms to separate the variables: x f^ (x) = (x^2 - 1) f(x) f^ (x) f(x) = x^2 - 1 x = x - 1 x Integrating both sides with respect to x : f^ (x) f(x) dx = (x - 1 x ) dx f(x) = x^2 2 - x + C Now, us