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Functions that are not integrable in terms of elementary functions.

by ╔(σ_σ)╝
Tags: elementary, functions, integrable, terms
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Sep27-09, 12:28 PM
╔(σ_σ)╝'s Avatar
P: 849
I was doing my differential equations homework. I had to solve y'' -4y = (e^(2x))/x.

While doing this I ran into an integral[tex]\int\frac{e^{4x}}{4x}dx[/tex]. I tried integrating my times but I couldn't; my guess is that this cannot be integrated in terms of elementary functions but I'm not sure.

Is there a theorem or Algorithm for knowing if a function is integrable in terms of elementary functions or not ?

If so, can someone tell me the theorem ?

And in my case is my function integrable in a finite number of elementary functions ?
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Sep27-09, 12:54 PM
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Hurkyl's Avatar
P: 16,091
I believe there is a theorem, although it's not particularly simple.

As a practical matter, I've found mathematica a good enough test.
Sep27-09, 01:45 PM
╔(σ_σ)╝'s Avatar
P: 849
Yeah, I saw the answer given by mathematica after my post.

But do you have a link to the theorem
I would like to see it even though, I may understand little of it.

Sep27-09, 03:07 PM
P: 607
Functions that are not integrable in terms of elementary functions.

R. Risch, The problem of integration in finite terms, Trans. Amer. Math. Soc , 139 (1969), 167-189. Mathematical Reviews (MathSciNet): MR38:5759

KASPER T. (1980): "Integration in Finite Terms: the Liouville Theory", Mathematics Magazine 53 pp 195 - 201.

Maxwell Rosenlicht, Pacific Journal of Mathematics 54 (1968) pp 153 - 161 and 65 (1976) pp 485 - 492.
Sep27-09, 05:36 PM
╔(σ_σ)╝'s Avatar
P: 849
Nov18-09, 10:33 PM
P: 13
Maybe you could expand e^{4x} in its Taylor series expansion. and then look at the integral of (e^{4x}-1)/x+1/x. In the first term there will be a cancellation of a power of x so it will be a polynomial integration. and the second gives ln(x). But depending on your integration bound there might be an issue with ln(x) (it blows up at x=0).

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