Michael Monagan, Simon Fraser University (5 lectures, 11 hours)
These are lectures on algorithms for computing an antiderivative ∫ f(x) dx where f(x) is a rational function of x or an elementary function of x.
Lecture 1 : Rational function integration: the rational part. Hermite reduction and Horrowitz's algorithm.
Video Lec21Anotes.pdf Lec21Bnotes.pdf Lec21Cnotes.pdf Lec21Handouts.zipLecture 2 : Rational function integration: the logarithmic part. The Trager-Rothstein resultant.
Video Lec22Anotes.pdf Lec22Bnotes.pdf Lec22Cnotes.pdf Lec22Dnotes.pdf Lec22Handouts.zipLecture 3 : Elementary function integration: the Risch structure theorem, Liouville's theorem.
Video Lec23Anotes.pdf Lec23Bnotes.pdf Lec23Cnotes.pdf Lec23Handouts.zipLecture 4 : The Risch algorithm: logarithmic extensions.
Video Lec24Anotes.pdf Lec24Bnotes.pdf Lec24Cnotes.pdf Lec24Handouts.zipLecture 5 : The Risch algorithm: exponential extensions.
Lecture 25 Lec25A.pdf Lec25B.pdf Lec25C.pdf Lec25Handouts.zipWhat I find interesting is that all of the algorithms boil down to polynomial arithmetic in one variable over a field F where, for example, F = ℚ or F = ℚ(x).
The tools used for rational function integration are (1) the Euclidean algorithm which is used to solve polynomial diophantine equations in F[x] and also for computing the square–free factorization of a polynomial in F[x], and (2) the (Sylvester) resultant of two polynomials in F[z][x], which, incidentally, can also be computed efficiently using the Euclidean algorithm.
For the Risch algorithm we use the same tools for polynomials in F[Θ] where F is a differential field (F is closed under differentiation), for example F = ℚ(x), and Θ is a differential extension of F, for example Θ = log x and Θ = ex. A more complicated example where the integrand f(x) involves both log x and ex would be F = ℚ(x,ex) and Θ = log x.