Functions that are not integrable in terms of elementary functions.by ╔(σ_σ)╝ Tags: elementary, functions, integrable, terms 

#1
Sep2709, 12:28 PM

P: 851

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 ? 



#2
Sep2709, 12:54 PM

Emeritus
Sci Advisor
PF Gold
P: 16,101

I believe there is a theorem, although it's not particularly simple.
As a practical matter, I've found mathematica a good enough test. 



#3
Sep2709, 01:45 PM

P: 851

Yeah, I saw the answer given by mathematica after my post.
But do you have a link to the theorem É I would like to see it even though, I may understand little of it. 



#4
Sep2709, 03:07 PM

P: 608

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), 167189. 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. http://en.wikipedia.org/wiki/Risch_algorithm 



#6
Nov1809, 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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