Let f:[1, ) R be a differentiable function satisfying ₁^x t^2 f(t) dt = x^3 f(x) - x^4 + 1 for all x 1 . If A is the area of the region bounded by the curve y = f(x) , the x -axis, and the lines x = 1 and x = 2 , then the value of 3A is :

Step-by-step solution

Given the integral equation: ₁^x t^2 f(t) dt = x^3 f(x) - x^4 + 1 Differentiating both sides with respect to x using the Newton-Leibniz formula: x^2 f(x) = 3x^2 f(x) + x^3 f'(x) - 4x^3 Since x 1 , we can divide the entire equation by x^2 : f(x) = 3f(x) + x f'(x) - 4x x f'(x) + 2f(x) = 4x f'(x) + 2 x f(x) = 4 This is a linear differential equation. The integrating factor (I.F.) is: e^ 2 x dx = e^ 2 x = x^2 Multiplying by the I.F. and integrating: f(x) x^2 = 4x^2 dx f(x) x^2 = 4 3 x^3 + C f(x) = 4 3 x + C x^2 To find