Question

Is f(x)=x^3 the only possible antiderivative of f(x)=3x^2? If not, why not?

Answer 1

No, it isn't.

If `f(x)=x^3` then the derivative will be `f'(x)=3x^2`

But the same would be true for `f(x)=x^3+1`
because the `1` would leave `0` in the derivative.

In general:
The antiderivative of `f'(x)=3x^2->f(x)=x^3+C`
(`C` being any number you choose)

This goes for all antiderivatives. You can always add `C`
(because they disappear in the other-way-around process)