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Why is differentiating mechanics and integration art?

There is a formula for the derivative of any product, composite or sum of functions, in terms of the derivatives of those functions. Just use the chain rule, product rule and sum rule.

But the same doesn't hold for integration.

Why not?

This is made even more surprising by the fact that differentiability implies integrability (many more functions are integrable than differentiable), and by the fact that (apparently) the derivative of a computable function from R to R might not be computable (assuming it exists), but the integral of a computable function from R to R must be computable (again provided that it exists).

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This was discussed as part of the older question mathoverflow.net/questions/66377/…. –  Ryan Reich Feb 28 '13 at 16:17
    
Voted to close as duplicate. I provide the link again as the earlier (inadvertently) links to a specific answer mathoverflow.net/questions/66377/… which is slightly inconvenient. –  quid Feb 28 '13 at 16:47
    
Oops, unintentional vanity. I found the question by searching my answers. –  Ryan Reich Feb 28 '13 at 22:09
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marked as duplicate by Andrej Bauer, Ryan Reich, quid, Qfwfq, Todd Trimble Feb 28 '13 at 17:25

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I think this question is not research-level, but nevertheless, since the answer is not generally known in the mathematical community, here it is.

You are mistaken. There is a "formula" for integration. It goes under the name "Risch algorithm". That is, there is a completely mechanical procedure which computes the closed form of an indefinite integral of a function which is a composition of elementary functions, if there is one and fails otherwise. However, the procedure is quite a bit more complicated, and is thus not generally taught in schools. In fact, it is so complicated that complete implementations are hard to come by. Caveat: there are parts of Risch's algorithm which are not known to be entirely algorithmic (see Wikipedia for a discussion of this point).

The general theory and problem of which the Risch algorithm is the solution was developed already by Liouville.

There is also a more simplistic answer. Many rules of differentiation can be inverted to give corresponding rules of integration, for example:

  1. The product rule becomes integration by parts.
  2. The chain rule becomes the substitution rule.
  3. Integral of $x^n$ is deduced from the derivative of $x^{n+1}$.
  4. Integral of $\sin x$ is deduced from the derivative of $\cos x$, etc.

The trouble is, while the rules of differentiation provide a complete set of rules for getting rid of all the derivatives in a systematic fashion, the corresponding rules for integration do not. So many think that integration is an art. It isn't, it's an algorithm invented by Risch.

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It is an overstatement to call the Risch process an algorithm in the traditional sense (even the wikipedia page says why: it involves zero-testing for composites of elementary functions). Also, you said that the rules of differentiation provide a complete set if rules for getting rid of all the integrals... surely you mean differentials? Nice answer otherwise! –  Vidit Nanda Feb 28 '13 at 16:18
    
Thanks, I fixed the mistake and put in a caveat about Risch's algorithm being only an "algorithm". –  Andrej Bauer Feb 28 '13 at 18:42
    
Thanks, Andrej. My post is a duplicate question. The original question has a pretty awful title - I searched for an answer before posting and didn't realize that the other title "Why is differentiating mechanics and integration art?" meant the same thing as my question. –  Rationalist Feb 28 '13 at 19:28
    
As for the Risch process, it seems to rely on special facts about a certain restricted class of functions (elementary functions) so isn't anywhere near an equivalent to the product and chain rules. –  Rationalist Feb 28 '13 at 19:30
    
I disagree, I think the theory behind it is a bit deeper than just "special facts about certain restricted class of functions". –  Andrej Bauer Feb 28 '13 at 23:43
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