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Has it been proven that there is no formula for analytic integration? Is this still open, could there be a formula?

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What do you mean? Do you mean an algorithm for symbolic integration? There is one that almost works: en.wikipedia.org/wiki/Risch_algorithm –  Potato Jun 15 '12 at 9:07
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I am not sure what the question is. –  Eric Naslund Jun 15 '12 at 9:26
    
a reference: Integration in Finite Terms. Maxwell Rosenlicht. The American Mathematical Monthly, Vol. 79, No. 9 (Nov., 1972), 963-972. –  GEdgar Jun 15 '12 at 15:38
    
@EricNaslund: I believe that the OP is trying to find if there is a general formula or method that can be applied to a wide variety of integrals. I was wondering something similar. I started with a product of two functions of $x$, and tried to work backwords to find the integrals and sums that would create them. I thought that I may be able to find a method that could start with a wide variety of functions, and convert them between integrals and summations. This seems to hint that this is very hard... –  Matt Groff Jul 17 '12 at 22:23
    
I basically want to know if there is a proof that there is no such formula similar to the differentiation formula for analytic integration. –  909 Niklas Jul 19 '12 at 5:16

1 Answer 1

up vote 1 down vote accepted

I assume the question is "differentiation has formulas that you can just plug functions into to get your answer. Do these exist for integration?"

In which case the answer is no, though as previously stated the Risch algorithm gets you most of the cases where they do exist. Try solving $\int \sin(x^2)$, for example (but don't try too hard).

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The example Robert gave is not expressible in terms of elementary functions, so new functions had to be conjured to represent the result of this integral (the Fresnel integrals). –  J. M. Jun 15 '12 at 17:08
    
Thank you very much for the answer. –  909 Niklas Jul 19 '12 at 5:17

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