Can you suggest advanced books on integral calculation?

I could suggest the following

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.

Besides the suggestions below, it is instructive to go to the math stackexchange and look at the highest voted Q&As tagged 'integrals' or 'definite-integrals'. Legends like Cleo lurk there demonstrating mind-boggling feats of integration. The corresponding mathoverflow tags are also interesting, but likely to involve more advanced concepts.

As practical advice (not addressing your enjoyment), learn how to use a CAS effectively. That will have a ton more bang for the buck in studying physics than being able to occasionally impress yourself with a Feynman trick or obscure application of the King property.

Along a similar line, the only book that I've found that has goes through a careful derivation of the divergence theorem using Lebesgue integration is, "A Concise Introduction to the Theory of Integration," Second Edition by Daniel W. Stroock. I prefer this one to his later book, "Essentials of Integration Theory for Analysis."

Does anyone else have a good reference or book on the topic?

"A Course of Higher Mathematics" - Smirnov

"Differential and Integral Calculus" - Piskunov

"Problems in Mathematical Analysis" - Demidovich (exercises/problems)

Advanced Calculus by Woods.

https://zackyzz.github.io/feynman.html

"Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals" - Boros and Moll

"Inside Interesting Integrals" - Nahin

Learn complex analysis contour integrals! There super fun for doing physics integrals. I'm not sure if these are too basic for what your asking about or not, but thought I'd mention them. Junjiro Noguchi's Introduction to Complex Analysis is a book recommended from herehttps://math.stackexchange.com/questions/438468/what-is-the-...

It's fantastic that you enjoy tackling challenging integrals, especially as you embark on your master's in physics! To master advanced techniques, consider exploring resources like "Advanced Calculus" by Patrick M. Fitzpatrick, or delving into specific areas like contour integration for complex analysis.

Since you are already doing this, can you suggest some good books you have be using for beginners. ?

Get Gradshteyn and Ryzhik and follow the references ...

https://en.wikipedia.org/wiki/Gradshteyn_and_Ryzhik

It might be a bit disappointing, but there aren’t much in the way of next level techniques.

Sure there might be one off tricks to solve specific problems, but for integration these tend not to be useful in general.

The MIT integration bee has some good problems. So do the Putnam tests.

I remember study guides for the first actuarial exam give tricks to calculate repeated integrals fast.

There's also those CRC manuals filled with solutions to integrals that show the steps.