Are there human "integrators", "differentiators", or "analysts"? I've heard of and seen people capable of performing seemingly complex arithmetic calculations mentally. Often this involves memorizing an algorithm which simplifies the process somewhat, for example, calculating squares of two or three digit numbers.
Are there any individuals that are able to mentally perform integrals or derivatives requiring multiple steps? How about working through a real analysis proof mentally? I.e. is there a person capable of performing a type of calculation that can't really be done via a standard algorithm but simply requires keeping track of multiple abstract or complicated objects mentally?
I suppose it might be likely that there is a person out there who has memorized, say, the 3 pages of integral tables in the back of a calculus textbook. I would consider that impressive no doubt!
An example: is there someone who do the integral: $$\int \frac{1}{x^2\sqrt{x^2+1}} \ dx$$ mentally? This may not be too hard of an example, and some practice with trig substitution may even lead someone to having the answer memorized more or less. However, how about a trig substitution which is quite a bit more complicated, where one works through the calculation mentally, step-by-step? Are there any "public performers" or users of this site who do that?
Are there any references out there which offer tips or tricks for performing standard integrals and derivatives mentally?
