# Functions that are not integrable in terms of elementary functions.

1. Sep 27, 2009

### ╔(σ_σ)╝

I was doing my differential equations homework. I had to solve y'' -4y = (e^(2x))/x.

While doing this I ran into an integral$$\int\frac{e^{4x}}{4x}dx$$. 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. Sep 27, 2009

### Hurkyl

Staff Emeritus
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. Sep 27, 2009

### ╔(σ_σ)╝

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. Sep 27, 2009

### g_edgar

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.

http://en.wikipedia.org/wiki/Risch_algorithm

5. Sep 27, 2009

### ╔(σ_σ)╝

Thanks.

6. Nov 18, 2009

### NSAC

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).