We thus often write
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
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