Home | 18.01 | Chapter 11

Tools    Index    Up    Previous    Next

11.3 Uniqueness of Antiderivatives

Suppose A(x) and B(x) are two different antiderivatives of f(x) on some interval [a, b].



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 .