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The elementary real-valued functions are not closed under integration. (Elementary function has a precise definition -- see Risch algorithm in Wikipedia). This means there are elementary functions whose integrals are not elementary. So we can construct a larger class of functions by adjoining all the integrals of elementary functions. You can repeat this process indefinitely. If I understand things correctly, the set of functions that is the countable closure of this process is closed under integration. Does any finite iteration of the process achieve closure under integration?

My guess is no. Has anyone thought about this?

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    Probably people who work in differential Galois theory have thought about this; someone familiar with the field should be able to give a straight answer.2010-08-22
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    Probably somebody who can explain this paper: http://projecteuclid.org/euclid.pjm/1102104969 to lesser beings like myself have a chance at resolving this question to everybody's satisfaction.2010-08-23
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    Does the elementary integral of an elementary function always have an elementary integral?2013-04-09
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    I don't know what you mean by "elementary integral". If you mean "if the integral f of a given elementary function g is also elementary, is f elementary? The answer is no: e^(x^2) is the integral of the function 2x e^(x^2), but e^(x^2) is not elementary.2013-04-10
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    See also http://math.stackexchange.com/questions/474034/families-of-functions-closed-under-integration.2014-06-16

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