## Indefinite Integrals

The notation means that on some (usually specified) domain of definition of .

Definition 2.2.1 (Anti-derivative)   We call an anti-derivative of .

Proposition 2.2.2   Suppose is a continuous function on an interval . Then any two antiderivatives differ by a constant.

Proof. If and are both antiderivatives of a function , then Thus from some constant (since only constant functions have slope 0 everywhere). Thus as claimed. We thus often write where is an (unspecified fixed) constant.

Note that the proposition need not be true if is not defined on a whole interval. For example, is not defined at 0. For any pair of constants , , the function satisfies for all . We often still just write anyways, meaning that this formula is supposed to hold only on one of the intervals on which is defined (e.g., on or ).

We pause to emphasize the notation difference between definite and indefinite integration.  a specific number  a (family of) functions

One of the main goals of this course is to help you to get really good at computing for various functions . It is useful to memorize a table of examples (see, e.g., page 406 of Stewart), since often the trick to integration is to relate a given integral to a known one. Integration is like solving a puzzle or playing a game, and often you win by moving into a position where you know how to defeat your opponent, e.g., relating your integral to integrals that you already know how to do. If you know how to do a basic collection of integrals, it will be easier for you to see how to get to a known integral from an unknown one.

Whenever you successfully compute , then you've constructed a mathematical gadget that allows you to very quickly compute for any (in the interval of definition of ). The gadget is . This is really powerful.

William Stein 2006-03-15