In the next four hours I will introduce you to the Risch Integration Algorithm. This is in some sense the highlight of the course, so I will be taking some time with it. In today's two hours, I prepare the groundwork by showing you how a CAS integrates simple rational functions. In the first hour I review the ordinary partial fraction algorithm that you all know, and discuss its liabilities from a computer algebra point of view. In the second hour today I will discuss modern (i.e. only at most a hundred years old) improvements, which will turn out to generalize nicely to the general elementary function case. Next week we'll formally define elementary functions and discuss the Risch Integration Algorithm proper.
These lectures will be more mathematical than any of the previous lectures. I feel somehow as if I should apologise to the physicists and other non-mathematicians in the audience for this, which is peculiar as this is a math class, after all. But I won't apologise, because I feel that the mathematics I will be discussing today and next week has real value. If you learn the mathematics behind the algorithms you will understand Maple's results (and sometimes non-results) better, and be able to use Maple better, not to mention with more confidence. But the real reason that I am including some details of the mathematics here is simply that it is powerful, elegant, and attractive, and I hope that after seeing it you agree.