




Suppose A(x) and B(x) are two different antiderivatives of f(x) on some interval [a, b].
Then
.
According to our Corollary to the mean value theorem in the last chapter, this implies A(x)  B(x) is a constant on the interval [a, b], which can be written as
A(x) = B(x) + c on [a, b].
Thus any two antiderivative of the same function on any interval, can differ only by a constant. The antiderivative is therefore not unique, but is "unique up to a constant".
The square root of 4 is not unique; but it is unique up to a sign: we can write it as 2.
Similarly, the antiderivative of x is unique up to a constant; we can write it as .