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I noticed that some users here are astonishingly skilled at evaluating integrals and I'm not talking about integrals you can find in Calculus II that can be solved with basic/traditional techniques (e.g., Trigonometric Substitution and Partial Fraction Decomposition), I'm talking about these kind of integrals as well as these.

So my questions are:

  1. What are the books, sources, studies, or websites you utilized to attain such a high level of mastery in solving integrals?

  2. How do you study new integration techniques? Do you just focus on a specific technique until you mastered it? Or do you learn new techniques by stumbling upon a specific integral that you can't evaluate using your current toolbox?

  3. How do you know which integration technique to use on a specific integral? How do you know so many special functions and where (and when) to utilize them in an integral?

I apologize if I have so many questions, this is my first post on this website. I am just amazed to see users destroy very hard integrals like it's some sixth grade algebra problem.

Jaiedeline
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    Some of it comes from the courses beyond calculus, such as real and complex analysis, Fourier analysis, etc. Some techniques and "tricks" are discussed in books like (recent ones) Correct Anti-differentiation: The Change of Variable Done Well by Martinez-Abejon, Inside Interesting Integrals, Nahin; (Almost) Impossible Integrals, Sums, and Series, Valean; as well as older compliations (many available on-line). And many of the solvers have a great deal of knowledge and experience. –  Sep 19 '22 at 06:02
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    That second integral somewhat reminds me of this beautiful result. – Christian E. Ramirez Sep 19 '22 at 06:12
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    Most of those skilled integrators you are referring to developed their abilities right here on this site over a period of years, learning from each other’s insights and improving upon them. I’ve been here nearly a decade and witnessed the growth of many users in real time! – David H Sep 19 '22 at 06:12
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    Don't feel disheartened that your question was closed (it seems well thought out). That question you were linked has quite the reputation here, and has some pretty informative and amazing answers by some people who are really skilled at integration. It should be a good resource for you. – Christian E. Ramirez Sep 19 '22 at 06:23
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    Those "skills" are just the application of the right method (or shall we say trick) on the right time. The users apparently learnt such tricks intensive and therefore can calculate many "impossible looking" integrals. – Peter Sep 19 '22 at 13:37

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