Let a differentiable function f(x) satisfy x^2 f(x) + ₁^x t f(t) dt = 4 5 x^5 for x 1 . Then the value of 20 ₁^2 f(x) dx is equal to:

Step-by-step solution

Given equation: x^2 f(x) + ₁^x t f(t) dt = 4 5 x^5 Differentiating both sides with respect to x using the product rule and the Newton-Leibniz formula: 2x f(x) + x^2 f'(x) + x f(x) = 4x^4 x^2 f'(x) + 3x f(x) = 4x^4 Dividing by x^2 (since x 1 ): f'(x) + 3 x f(x) = 4x^2 This is a first-order linear differential equation. The integrating factor (I.F.) is: I.F. = e^ 3 x dx = e^ 3 x = x^3 Multiplying the equation by x^3 : x^3 f'(x) + 3x^2 f(x) = 4x^5 d dx (x^3 f(x)) = 4x^5 Integrating both sides with respect to x : x^3 f