If f'(x) e^ x^2 ,dx = (x - 1) e^ x^2 + k , where k is constant of integration, then f(x) =

Step-by-step solution

Differentiating both sides of the given equation with respect to x : d dx [ f'(x) e^ x^2 ,dx ] = d dx [ (x - 1) e^ x^2 + k ] f'(x) e^ x^2 = 1 e^ x^2 + (x - 1) e^ x^2 (2x) f'(x) e^ x^2 = e^ x^2 (1 + 2x^2 - 2x) Dividing both sides by e^ x^2 : f'(x) = 2x^2 - 2x + 1 Integrating both sides with respect to x to find f(x) : f(x) = (2x^2 - 2x + 1) ,dx f(x) = 2x^3 3 - x^2 + x + c where c is the constant of integration. Answer: 2x^3 3 - x^2 + x + c , where c is constant of integration.