Inside Interesting Integrals, by Nahin. This contains integrals with often arithmetic inspiration. Imagine things like, lots of special values of the zeta function or partial sums of harmonic numbers and that sort of thing.
Irresistible Integrals, by Boros and Moll. This is inspired by their work to prove all of the integrals in the enormous book of Gradshteyn and Ryzhik, whose references are often old and lacking (like Erdelyi's prior table of integrals).
You might also like Solved Problems for the Gamma and Beta Functions, Legendre Polynomials, and Bessel functions by Farrel and Ross. This is a bit closer to real analysis and veers towards more practical integral magic.
Finally, I'll note that all of these largely omit complex residue calculus (using complex analysis to solve real integrals). I don't know of a good book specifically aimed at this, unfortunately.
(Almost) Impossible Integrals, Sums, and Series - Cornel Ioan Vălean
Does anyone else have a good reference or book on the topic?
To me, part of the value of Lebesgue integration is in understanding the limitations of Riemann integrals and when they break. Some of this is covered in Stroock's book in chapter 5.1. Alternatively, when in working in function spaces, we may need to integrate in a more general way than Lebesgue integration, so things like Bochner integrals, which require similar theory. This can arise in the theory related to things like PDE constrained optimization, which most of the time is targeted toward physics related models.
All that said, bluntly, I prefer to work with Riemann integrals when at all possible. However, the same question then applies. Do you or someone else have a reference for a rigorous derivation of the divergence theorem or integration by parts in multiple dimensions using Riemann integration? It's not particularly hard in one dimension, but higher dimensions is tricky and it's hard to get the details of integrating on the surface correct. Stroock's book is the only reference that I know of and he does it with Lebesgue integration.
The more practical advantage measure theory provides for probability is you can simultaneously handle continuous and discrete distributions. Most of the time what works for one works for the other but you can get some weird mistakes (Shannon’s differential entropy has a few issues as a measure of information not found in the discrete case because he got lazy and just replaced the sum with an integral).
A good chunk of the time I come across measure theoretic probability papers I feel like they’re making the paper a lot more complicated and messier than it needs to be, but it does serve a purpose.
"Differential and Integral Calculus" - Piskunov
"Problems in Mathematical Analysis" - Demidovich (exercises/problems)
"Inside Interesting Integrals" - Nahin
Sure there might be one off tricks to solve specific problems, but for integration these tend not to be useful in general.
There is a widely applicable generic algorithm for indefinite integration (https://en.wikipedia.org/wiki/Risch_algorithm).
> but for integration these tend not to be useful in general.
That is true. For many problems, simpler methods work.
Also, according to that page “Currently, there is no known full implementation of the Risch algorithm”.
There's also those CRC manuals filled with solutions to integrals that show the steps.
