Let be an open interval of and a real function.

A function is called an antiderivative of if is differentiable and its derivative is equal to , i.e.

   for all

Note that there are an infinite number of antiderivatives for any function since any constant can be added or subtracted from any valid antiderivative to yield another equally valid antiderivative.

To account for this, we express the general antiderivative, or indefinite integral, as follows:

There is no loss in generality with this notation since in fact all antiderivatives of take this form as the following theorem demonstrates:

Theorem. Let be two antiderivatives of a given function defined on an open interval . Then .

Proof. Since and , we have . Thus, .

This is no longer true if the domain of the function is not an open interval (is not connected). For that scenario, the following more general result holds:

Theorem. Let be an open set (not necessarily an interval). Suppose are antiderivatives of a given function . Then is constant in each connected component of (each interval in ).

For example, consider the function given by . Notice that the domain of is not an interval, but the union of the disjoint intervals and . Then, all the antiderivatives of take the form

Remarks

• For complex functions, the definition of antiderivative is exactly the same and the above results also hold (one just needs to consider “connected open subsets” instead of “open intervals”).