How many integration techniques are worth knowing?

[MathWarrior]

I would say you need to know all the integration techniques to some degree, basic ones you encounter in calculus 1-3 definitely. This however depends entirely on what your major is, and what you intend to do.

 

[cgk]

Some guys in this forums found integrals that Mathematica couldn't compute: http://www.wilmott.com/messageview.cfm?catid=10&threadid=85389

That's not hard to come up with. Just write down a horrible contrived expression, take its derivative, and--tada--you got an integral you know the answer to but no one else (human or machine) has any chance of ever calculating.

The question is not whether there are expressions for which a--possibly simple--closed-form analytical expression of its integral exists. Of course there are. The question is: If you do not happen to be able to already know or be able to guess the answer, do you, as a human, have any chance of coming up with an algorithm to get the answer which Mathematica cannot? And this can usually happen only if you have some additional information about the problem (e.g., symmetries, magic coordinate transformation, etc) which Mathematica does not have. Because it sure *DOES* know how to apply all the basic integration techniques, and it does know them much better than you as a human do.

Although I have do admit, I was quite surprised by the BesselJ example. I'll try this some time (maybe some assumptions were missing, and the posted identity doesn't always hold).

btw: I'd bet several of the examples on that page would just work in Mathematica if you turned on the cubic and quartic radicals (which are disabled by default because calculating with them might stress your patience). And the fact that you can just do something like this: Ask the program to solve with techniques you could never hope to apply by hand, is already quite awesome. (actually, most of the symbolic integration techniques one could never apply by hand. Some of them are really cool.)