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

Jan 27, 2015

No, it isn't.

If $f \left(x\right) = {x}^{3}$ then the derivative will be $f ' \left(x\right) = 3 {x}^{2}$

But the same would be true for $f \left(x\right) = {x}^{3} + 1$
because the $1$ would leave $0$ in the derivative.

In general:
The antiderivative of $f ' \left(x\right) = 3 {x}^{2} \to f \left(x\right) = {x}^{3} + C$
($C$ being any number you choose)

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