## Personal Background

I am by no means any expert at integration, but I have done a fair share of it. Integration is like a hobby to me. Instead of solving puzzles or riddles, I love evaluating integrals. As an example I am in the process of writing some personal notes on integration. Hopefully at least the questions will be translated not too long into the future.

As well as writing those notes I have a bachelor degree from the university. Here I have taken as many related topics to integration as possible. Analytic Number Theory, Quantum Mechanics, Complex analysis, etc.

Many claims that integration is an art. To continue this analogy how does one become a master of "art"? Naturally the easiest way is to be a natural born genius like Leonardo Da Vinci (or Ramanujan). Luckily, this is not the only way. If you want to become good at something you have to train hard, in order to master something you have to work ten times harder than that.

### Main steps

Below are the main steps (IMHO) on the road towards being a integration god/goddess.

- Read books, papers, journals etc about integration.
- Read books, papers on different areas of mathematics.
- Solve a ridiculous amount of problems

Which basically boils down to find interesting integrals, solve them and repeat, repeat, repeat ad absurdum.

## Specific books and papers

### Books about integration

(*table of integrals, series and products, Gradshteyn and Ryzhik*)
It can be cumbersome to look up at times, but it has an insane amount of formulas in there. The book is constantly being revised for errors and new integrals are added at every new edition. The digital searchable version is in my opinion much better.

(*Irresistible Integrals, Boros and Moll*) Regarded as the holy grail of symbolic integration among the many book. I think it is a fine book, but not my personal favorite. It is a bit cluttered and hard to navigate. I much prefer the excellent articles by Victor Moll instead.

(*Handbook of integration, Daniel Zwillinger*) I am quite fond of this book. The book is perhaps a tad short, but it makes up for it by being very handy and has a wild amount of further references.

(*Inside interesting integrals*, Nahin) Not a favorite of mine. Although I think many others will enjoy it. It tries to explain the reason behind integration and how to come up with the various techniques and skills. It does this well, and the book has a decent amount of problems the reader is strongly encouraged to solve throughout the book. Even though I had seen most of the problems before I liked the references and some of the ideas that were discussed.

(*The Gamma Function, Artin*) I use this book somewhat like a thesaurus. If there is some property of the gamma function or a proof I need to study I just look here. I liked the historical background and the written out proofs. Sometime these can be a bit too wordy, but the presentation of the Bohr-Mullerup theorem is much clearer presented than in for example Baby Rudin.

### Papers

Compared to books there are literarly hundreds of interesting articles about integration. I will just mention a handful here

(*The Gaussian Integral, Keith Conrad*). Exactly what is says on the can. A collection of proofs for the classical $\int_{\mathbb{R}}e^{-x^2}\,\mathrm{d}x$ integral.

The next part is more a collection of articles. Look at the papers written by Victor Moll in collaboration with various others. As an example he has written an article on a class of logarithmic integrals. I really liked his series of articles regarding proving formulas from (*Gradshteyn and Ryzhik*). See for an example $R(x) \log x$ or $\psi(x)$. It seems he have removed these articles from his home page, in preporation for his book. Which I must say I am looking forward to.

(The Gamma Function, 0504432) This is a nice little paper written by a student. The similarity between this paper and the book "*The Gamma function*" is striking. However if one does not want to read a whole book this is a clear and concise presentation.
*Note:* Both the book and this article skimps out on some details. Some proofs are omitted and others are not clear enough. To get some deeper results one can for example read the following paper.

(*"Advanced" Integration Techniques, Michael Dougherty*) I took the liberty of adding quotation marks around advanced. This is a simple and clean treatise of the basic integration techniques. What really struck me was the pitfalls of substitutions and the clear presentation of trigonometric substitutions. This is perhaps way below what you are looking for. However I found it a worthwhile read even if I knew the theory beforehand.

There are many other good articles. Some on derivation under the integral sign, while others spend 30 pages on the history behind the gamma function.

### Websites

Stack exchange has many well formulated questions about integrals, however there are other websites as well.

(mathematica.gr) A useful site for bears and comrades. This thread in particular deals with definite integrals. This was the thread that helped me learn contour integration while studying complex analysis.

(Advanced integration techniques, Zaid Alyafeai) I wonder where everyone comes up with such original names. Anyway the thread is very nice. Takes a peak at some special functions and shows various ways to prove the Euler-reflection formula $\Gamma(z)\Gamma(1-z)=\pi/\sin \pi z$. I especially liked the proof with convolution.

### Books that contain interesting integration

Good books.. I liked Gamelin for Complex Analysis, Apostol for analytic number theory as well as (*The Theory of the Riemann Zeta-Function, Titchmarsh*)

(Introduction to Quantum Mechanics, Griffiths) Regarded as one of the best introdction books to QM. Also it is filled to the brim with more or less interesting integrals. It also introduces Lagrange, Laguerre and Bessel polynomials. Also the book is written in a wonderful and personal way. Truly a book a mathematician can read for joy.

### Ridiculous amount of problems

(Integral Kokeboken, Nebuchadnezzar) I know it is a bit weird to cite myself. However these notes contain a large amount of clever and cute integrals. After solving every problem in the intermediate section one is perhaps not yet a "master-integrator" but at least one has take a leap forward.

(Integration Bee) Again perhaps a tad too low level. These are bullet integration problems. Some are deviously hard, but others can be solved in seconds. The idea is you are given 20 problems and 30 minutes to solve as many as you can.

(Art of Problem Solving) This website was better before but still holds a fair number of interesting integration problems. In particular the user kunny a.k.a. Kunihiko Chikaya has a giant list of Japanese college problems. Note that most of these problems are either about integration or some kind of clever derivation.