Let be an open interval of and a real function.
To account for this, we express the general antiderivative, or indefinite integral, as follows:
where is an arbitrary constant called the constant of integration. The portion means “with respect to ”, because after all, our functions and are functions of .
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 on the whole . Thus, by the fundamental theorem of integral calculus, .
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:
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
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”).
|Date of creation||2013-03-22 12:14:55|
|Last modified on||2013-03-22 12:14:55|
|Last modified by||asteroid (17536)|
|Defines||constant of integration|