Where to learn advanced integral calculus?

I have learn the basic integration techniques: Integration by parts, partial fraction decomposition, recursion formulae, and others that would be considered in ordinary calculus courses. What amazes me on MSE is the plethora of methods to evaluate difficult integrals, may it be with Gamma or Beta functions, Frullani, different classes of integrals (elliptic, exponential etc.) and many others. I have yet to study residue calculus, neither Fourier analysis, but I would be surprised to see that these techniques would come up in those courses.

My question is if there is any literature on this matter, that covers these seemingly advanced techniques and not just the basic methods? Or is this just a matter of experience?

Solution 1:

I would recommend:

1. Inside Interesting Integrals by Paul Nahin. It contains many of the tricks that you mentioned.

2. The Handbook of Integration by Daniel Zwillinger. It is no longer in print but used copies can be found.

I have not read the following books but all are on my "wish list".

3. Integration for Engineers and Scientists by William Squire. It is cited repeatedly by Zwillinger. It is out of print but used copies are available.

4. Integral Evaluations Using the Gamma and Beta Functions and Elliptic Integrals in Engineering: A Self-Study Approach by C.C. Maican

While Nahin covers basic contour integration and part of Zwillinger uses contour integration, all of these books are suitable for those who have yet to study complex analysis.

Addendum

One of the members here, @Zaid Alyafeai, has written a freely available book on integration methods. I am currently looking at it and it is in the same vein as the books listed above. The link to the book can be found here.

Another good free resource: Integral Kokeboken (Integral Cookbook [Norwegian]).

Solution 2:

There are many, many different integration techniques that are available, depending on the difficulty of the integral involved and the level of the student. The more advanced methods you mention would probably only be needed as a post-graduate, given that Fourier and similar methods are used in 2nd year undergraduate studies.

Solution 3:

Take a look at this possibility:

Mathematics of Classical and Quantum Physics; by Frederick W. Byron, Jr. and Robert W. Fuller; (two volumes bound as one)

Solution 4:

You cannot go past these two recent texts, particularly if you are interested in certain types of integrals containing a logarithmic term.

1. (Almost) Impossible Integrals, Sums, and Series by Cornel Ioan Vălean (Springer: 2019).

2. An Introduction to the Harmonic Series and Logarithmic Integrals: For High School Students up to Researchers by Ali Shadhar Olaikhan (2021).

Both books deal extensively with a certain type of sum known as an Euler sum or its derivatives (alternating, non-linear, skewed, etc).