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How can you prove that a function has no closed form integral?

Since I was a college student, I was told there were many functions that cannot be integrated as simple functions.(I'll give the definition of simple functions at the end of the article). As a TA for calculus now, I've been asked for integrated various functions, certainly, most of them are integrable(in the sense of simple functions). However, how could I know that certain functions are not integrable not merely because I cannot integrate them. (There was one time that one integration on the question sheet daunted all the TAs I asked).

Does anyone know THEORETICAL REASONS why certain functions cannot be integrated as simple funtions? Or could you refer to certain reference containing such materials? Or, could you show me by example, certain "good" function actually don't have "good" integration, I think one famous example could be "$\frac{\sin x}{x}$".

Simple functions: The functions which is the summations(subtractions),multiplications(divisions), and compositions of the following functions(as well as the functions generated by these operations): $x, \sin x , \cos x, \log x, \sqrt[n]{x}, e^x$.

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marked as duplicate by J. M., Gerry Myerson, Argon, Zhen Lin, MJD Jul 10 '12 at 13:14

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

    
Relevant en.wikipedia.org/wiki/Elementary_function –  user2468 Jul 10 '12 at 0:10
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Seems to be a dupe of this question. Here is a related question. –  J. M. Jul 10 '12 at 0:12
    
Does $\log$ count as a "simply function"? You may be interested in as well: en.wikipedia.org/wiki/… –  Argon Jul 10 '12 at 0:16
    
Yes, I have edited the article. –  Li Zhan Jul 10 '12 at 0:19
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3 Answers 3

up vote 4 down vote accepted

A search for "integration in finite terms" will get you many useful results. This paper by Rosenlicht is a very good place to start.

Bibliographic details: Maxwell Rosenlicht, Integration in Finite Terms, American Mathematical Monthly 79 (1972) 963-972.

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Thank you so much! That is just what I am looking for! –  Li Zhan Jul 10 '12 at 0:22
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see http://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra) Liouville's theorem which, by wikipedia, places an important restriction on antiderivatives that can be expressed as elementary functions

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I can refer you to my website. I have been working on finding formulas for the $n$-th derivative and the $n$-th anti-derivative for elementary and special functions for many years.

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Your answer would be much more useful if you'd give a summary of what portions of the things in your website are relevant to this question. –  J. M. Jul 10 '12 at 4:00
    
@J>M.: Let people put some effort to gain the knowledge. Do not forget the saying "Easy come easy go". –  Mhenni Benghorbal Jan 25 at 20:33
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