Where to learn integration techniques?

Solution 1:

Where to learn integration techniques?

In college. $($Math, physics, engineering, etc$)$.

Is there any book that let you learn integration techniques?

Yes: college books.$($Math, physics, engineering, etc$)$.

I'm talking at ones like in this question here

That question does not require any fancy integration techniques, but merely exploiting the basic properties of some good old fashioned elementary functions.

the ones used by Ron Gordon

User Ron Gordon always uses the same complex integration technique, based on contour integrals exploiting Cauchy's integral formula and his famous residue theorem. They are pretty standard and are taught in college.

or the user Chris's sis or Integrals or robjohn.

See “Ron Gordon”. Also, familiarizing oneself with the properties of certain special functions, like the Gamma, Beta and Zeta functions, Wallis and Fresnel integrals, polylogarithms, hypergeometric series, etc. would probably not be such a bad idea either. In fact, there's an entire site about them.

Other users to watch out for are Achille Hui, sos$440$, Felix Marin, Random Variable, Tunk Fey, Vladimir Reshetnikov, Kirill, Pranav Arora, Cleo, Integrals and Series, Laila Podlesny, Olivier Oloa, etc.

Solution 2:

I think you are looking for some bridge between basic calculus and the very hard problems here done by gods like Ron and Cleo. I suggest looking at this old book. It actually has tidbits of very advanced techniques. But I think, because it still contains the standard techniques, there will be enough familiar to keep you comfortable.

https://archive.org/details/treatiseonintegr01edwauoft/page/n8

Other than that if you don't know complex analysis, I suggest that. You probably benefit more from a course that is more applied or calculational versus one with emphasis on proofs. A very easy entry to the subject is via Schaum's outline if you are self studying.

In addition, I recommend working through an entire standard ODE and PDE book. You will become familiar with many special functions in this context. In standard intro courses, there is the problem that you don't really have time to cover all the topics in the books. So work through and do all the "starred" sections that are skipped in a short course. Also look at the later homework problems (like 25-30 of a list of 30) where you will see some Chebychev, Airy, etc. I recommend this for similar reason as Edwards. Just think you need something a little bridgier and more in the context of the familiar to build your skills.