Counter example of antiderivative.

Question

Is it true ∫f(x) · g(x) dx = [ ∫f(x) dx] · g(x) + f(x) · [ ∫g(x) dx] ? If it is not true provide a counterexample. I know its not true but what would be a counterexample?

Answer 1

integral ( f g) = integral (f) * g + f * integral (g)

f = 1 and g=x^2

integral (f) = x and integral (g) = (1/3)x^3

integral (fg) = integral(x^2) = (1/3)x^3

plugging into the equation:

(1/3)x^3 = x *x^2 + 1 * (1/3)x^3 = x^3 + (1/3)x^3 = (4/3)x^3

it is false and there is a counter example

the only value where it holds true is x=0, upon solving that equation