Looking through old problems, it is not difficult to see that some users are beyond incredible at computing integrals. It only took a couple seconds to dig up an example like this.

Especially in a world where most scientists compute their integrals numerically, I find it astounding that people exist that can solve these problems analytically. Sometimes it is a truly bizarre contour, some examples use auxiliary functions and differentiation under the integral sign in a way that feels more like integration by wizardry.

Some of these users seem to have mastered an incredibly large number of special functions to a level of fluency that is almost unimaginable to me. Manipulations involving the error function almost looks like the work of an infant compared to some of these seemingly casual manipulations involving the Airy function, Barnes G function, the Legendre chi function and many more. Every time I read a post by them, it turns into an exercise in reading about yet another special function I have never heard of.

My question is what sort of material do you have to study, or what kind of area do you have to work in to get so good? To me, it doesn't seem as obvious as saying something like, "oh, they know this material because they are a topologist".

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    oh, they know this material because they are a topologist – Shahar Apr 22 '14 at 23:39
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    I fear every answer you'll get can be condensed to one word: Practice. I'm hoping there will be answers which are as constructive as possible, though; ones which essentially give a guide to "good practicing", and resources of problems/techniques to get started with. – FireGarden Apr 22 '14 at 23:44
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    Looking forward to Ron's answer! Even Wolfram Alpha is jealous of him. – Prism Apr 22 '14 at 23:46
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    @FireGarden I agree and I am not looking for some panacea. But as far as I am aware, they do not give out PHDs in antiderivatives nor do big companies hire scientists just because they are good at calculating integrals. Many of the techniques feel so dehumanized, all the motivation behind having this knowledge has had all its traces erased to me. – JessicaK Apr 22 '14 at 23:47
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    @JessicaK I would imagine the motivation is entirely personal. There are probably a few people in this world with a burning passion for Peruvian nose flute music, but this certainly isn't externally motivated nor rewarded! I know of a number of people who get great satisfaction from evaluating difficult integrals! – FireGarden Apr 22 '14 at 23:52
up vote 76 down vote accepted

I hesitate to provide an answer to this because it feels very immodest, but perhaps I can provide something useful that doesn't merely restate the obvious or succumb to narcissism on my part.

First of all, I will state something kind of obvious, as @FireGarden has pointed out. How do you get to Carnegie Hall? Practice! The people who I consider to be integration gurus on this site (@sos440, @O.L, @RandomVariable, @robjohn, @achillehui, forgive me if I left anyone out) do not strike me as Ramanujan-like savants who get their inspiration from the goddess Namagiri. Rather, these are people who clearly 1) love working out integrals (and sums and related quantities), and 2) work out an enormous number of problems, completely and in great detail. So, yes, hard work and enthusiasm explain a lot.

This doesn't mean, however, that you sit with a textbook and merely work out the exercises in a boring, rote manner. What I mean by practice is the application of your knowledge to new, interesting problems. Find problems that look impossible to you right now. Maybe they are, but I guarantee you somebody somewhere has solved it. Here are some places to find interesting/challenging problems:

  • Putnam exam problem archives (start here); keep in mind that several Putnam exam problems are solved in Math.SE.
  • Actuarial P exam (tons of double integrals over interesting regions, as well as some interesting single integrals in the context of probability computations, here's a sample)
  • Graduate Physics texts (source of many integrals found here)

But practice is not the only thing. The really great aspect of a forum like SE is that we all learn from each other. So, as you look at these terrific solutions, don't worry about how the author came up with the idea. Worry instead about learning from the insight, and see if you can apply the lessons elsewhere. I for one can say that I have totally benefitted from reading through the solutions of my peers here. Nobody here is an island.

Do not be intimidated by special functions; you will learn about them as you need them. Special functions really are just high-falutin' definitions that one person or another has found useful at some time. They are only useful when they can be computed in far less operations than a numerical evaluation of the integral defining them. (I've ranted about this before.) They do not imply a level of mastery, but rather a level of familiarity with the art. Personally, I have no issue with answers expressed in terms of special functions as long as they represent the simplest closed form of a solution. Occasionally, one find poseurs who use fancy special functions when a much simpler form does the trick; usually they are exposed.

So what might you want to learn? In decreasing order of importance: 1) Calc II integration techniques (crucial to master), 2) Mathematica/Maple (for algebraic and numeric verification), 3) Ordinary differential equations (for exposure to basic special functions and series solution techniques), 4) Complex analysis (because the residue theorem is incredibly useful), 5) Integral transforms (Laplace, Fourier), although more for the cool integrals to do rather than any "techniques" taught, 6) Laplace's Method, Method of Steepest Descent, Watson's Lemma, and like asymptotic methods, 7) Statistical methods, especially the various PDFs used in various contexts. In some cases, knowledge of Lebesgue integration may be useful (although I plead guilty to inexperience there).

I hope this helps. I've likely left stuff out, so I may have more to say.

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    Among the list of gurus, you forget that user which write down the answer, walks away without any explanation and takes us a few weeks/months to reproduce his/her answers. – achille hui Apr 25 '14 at 22:42
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    To be fair, the most mind boggling integral machine on MSE is most likely that mystical user Cleo. It's too bad he/she doesn't tell us how he/she get his/her answer. – achille hui Apr 25 '14 at 22:57
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    @RandomVariable: actually, you epitomize everything I am talking about. I have seen you go from asking tons of basic questions about branch cuts and contour integration to using what you have learned to be a constant, consistently good provider of integration results. You are too humble. – Ron Gordon Apr 26 '14 at 1:26
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    Forgive me if I left anyone out. - Yes. You left yourself out. :-) – Lucian May 12 '14 at 14:08
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    yes, Cleo is so mysterious, I fancy he's someone working in Russian nuclear bomb department, that could only relax by doing integrals, and not allowed to provide more details... – athos Aug 1 '14 at 9:52

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.

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    The link for kunny is broken now. – Suzu Hirose Dec 13 '14 at 10:28
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    This is a really wonderful list, at least now I know what to do during my christmas break. – JessicaK Dec 13 '14 at 14:10
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    Definitely. Fantastic stuff, N3! – Kari Dec 13 '14 at 19:37
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    Very nice and a great supplement to Ron Gordons answer. Thanks for all these refs. +1 – Markus Scheuer Dec 15 '14 at 14:01
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    Damn, any english translation for your pdf? – galois Apr 22 '15 at 7:39

Well, I was surprised to see my name somewhere in this page. Anyways, I don't consider myself an integration guru but you may say inspired by them. Unfortunately, I don't have the same interest now as I did 4 years ago nevertheless, I will share my experience. I started my journey to learn about integration about 4 years ago in the second year of university. I thought I have some interest in integration and might be good to push my knowledge from the basics to the advanced. My first step was to register in a forum and see how things work. I would post an integration question in the forum and someone said use the gamma function. To be frank, I faced all kinds of problems to collect the material needed to understand a single special function. This was the start of the journey. I suffered a lot to to understand all the basics and to move form one function to another. Actually sometimes you might find books on some special functions like the one by Lewin on polylogarithms and on the other hand suffer to collect some information about other functions like the theta function, loggamma integrals etc. I would say my process was slow but it eventually got me some way. I would suggest for you or anyone reading this post to start by the following materials because they are easy to understand

  • Differentiation under the integral sign
  • Gamma function
  • Beta function
  • Digamma function
  • Polylogarithms

These are the easiest and you can find lots of materials on the internet.

Here is a Advanced integration techniques of my journey, it talks about all the special functions I mastered.

  • @DavidH, thanks. Just tried to make it easier for those interested in such materials. – Zaid Alyafeai Dec 9 '16 at 1:15

First of all, you don't need the models mentioned in the previous posts, you don't need to be like X,Y,Z, you have to be like you, to find yourself and your inner resources, to find the version of you forged with extremely hard work and passion. This is not easy, it might be a painful and very long road you have to assume, even going like that for years without getting the results you want.
That's a true fact you have to be aware of and accept it. I don't even mention that this way might seem such a nonsense for some of those around you, but you have to be extremely strong mentally and resist any attempt of being discouraged.

The proper mindset is to evolve indefinitely, without bounds (like the ones mentioned by Ron Gordon - by the way, why should one ever care the others options, limits?), and even Ramanujan,
if considered there as a limit, is there to be overpassed one day, and this performance would be a profound act of recognition and respect for all his work. Develop your mathematical thinking originally, no matter the problems you meet, try to find new ways, develop, create new ways, the process of creation is perhaps one of the most powerful ways to reach a deep understanding of the mathematics you do and for getting the results you expect and even more than that.

Then remember one of the critical points, you're never in a competition with anyone, you only want to acquire better and better knowledge, indefinitely, reaching the highest peaks absolutely freely, being independent of the results by others. Don't spend time looking around and trying to compare yourself with anybody, and don't be scared by the way some manage to post solutions here, behind all is just a huge amount of work, you don't have to be a genius to do anything of what has ever been posted here. I'm confindent that you can do anything and far more if you truly want it.

As a last word, challenge yourself permanently with much courage and high confidence, be tougher with yourself than others could be with you in terms of mathematical challenges, and be ready, prepared anytime to face the worst integrals. Never give up! When you fail, you retreat, build new weapons, new strategies, and return until you put everything down.

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    not concise but forthright and thorough. – Abr001am Sep 24 '16 at 22:12
  • @Agawa001 thanks – user 23571113 Sep 25 '16 at 13:18

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