# Functions cannot be integrated as simple functions [duplicate]

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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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–  user2468 Jul 10 '12 at 0:10
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

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

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
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.