Let f: R R be a differentiable function such that f(0) = 0 and x f(x) + 2x x^2 + 1 d x = f(x) x^2 + 1 + C where C is an arbitrary constant. Then the equation of the normal to the curve y = f(x) at x = 1 is

Step-by-step solution

Differentiating both sides of the given integral equation with respect to x , we get: x f(x) + 2x x^2 + 1 = d d x ( f(x) x^2 + 1 ) Applying the product rule on the right-hand side: x f(x) + 2x x^2 + 1 = f'(x) x^2+1 + f(x) ( x x^2+1 ) Multiplying the entire equation by x^2+1 yields: x f(x) + 2x = f'(x)(x^2+1) + x f(x) Canceling x f(x) from both sides gives: 2x = f'(x)(x^2+1) f'(x) = 2x x^2+1 Integrating both sides with respect to x : f(x) = (x^2+1) + K Using the initial condition f(0) = 0 : 0 = (1) + K K = 0 Thus, f