email@example.com schrieb: > > The Russian book "Integration of Functions" [...] published by A.F. > Timofeev [...] in 1948 provides [...] 533 integration examples [...]. > I believe the whole would make a good independent integration test > suite because the book antedates all automated integrators and because > the author claims to cover the field thoroughly [...]. > > There are the usual misprints in the book, but with both the integrand > and antiderivative available, the original meaning can be > reconstructed for all (or almost all) examples I think. Timofeev's > antiderivatives are usually close to the most compact form possible > (an exception is his consistent use of logarithms for inverse > hyperbolic functions), but they have to be checked for validity over > the complex plane, and be corrected if necessary [...]. Apart from > compactness, continuity (and reality) on the real axis might be worth > checking and repairing too. > > So, if 5 to 10 people were found willing to type in and check (and > perhaps correct or improve) 50 to 100 integrals and evaluations each, > a digitization of this corpus could be an almost pleasant task, and > surely quite useful. [...] >
Although no interest has been shown in this integration test suite, I have entered, checked, and amended all 81 integrals from chapter 1; the integrals are appended below. Occasional multiple evaluations illustrate alternatives worth pointing out, and integrals collected in vectors represent examples involving sub-cases. Suggestions how to format the integrals for easy testing of as systems as possible are welcome.
The antiderivatives hold on the entire complex plane and exhibit no artificial discontinuities on the real axis (assuming the function definitions of Derive). This does not apply to poles that cannot sensibly be integrated over; thus logarithmic poles introduce imaginary steps whose systematic removal via LN(x) <- 1/2*LN(x^2) would have been awkward in many cases.
On these integrals, the performance of Derive 6.10 has been measured with the integration variable limited to real values (the default mode around which the system was built) - to declare the variable complex should be considered unfair:
Derive 6.10 fails on examples 42 and 65; it automatically integrates example 29 only with the Trigonometry simplification flag set to Collect, which is not the default. When the integration variable is declared complex, Derive additionally fails on integrals 44, 64, and 81.
Perhaps other systems (and/or chapters) can be added here by and by. I am currently working on chapter 4 (132 integrals) in which I am expecially interested, but not expecting to finish this before some months have passed. I am not likely to enter the integrals from other chapters too (particularly not the remaining massive ones 2, 5, and 8).