How Can One Prove a Function Lacks an Elementary Antiderivative?

[TD]

It sounds a bit strange in English (although we have the term in Dutch), so I'm not sure whether this is a generally known/used mathematical term: 'primitivable'. If not, it just comes down to find a primitive function F of f, so that F'=f. Specifically, I'm wondering about primitive functions in terms of (a finite number of) elementary functions.

Recently, someone asked me to find

$$\int {x\tan xdx}$$

Using integration by parts, this comes down to finding

$$\int {\ln \left( {\cos x} \right)dx}$$

Because of this, and the result from Mathematica (involving poly logarithms), I told the guy that this function probably doesn't have a primitive function, at least not expressed as a finite number of elementary functions. I answer the same thing when someone asks to find the indefinite integral of the more known e^(-x²).

Now on http://mathworld.wolfram.com/IndefiniteIntegral.html" , it says that this is exactly the case for a few of those well-known integrals (numbered line 8). But what interests me is that it also states that Liouville was able to show this. I was wondering how one can show/prove such a thing, in general or for these specific functions. I wasn't able to find anything while searching, perhaps because I'm searching on the wrong terms.

Perhaps someone knows how this is done, can be done or where it can be found? I'd love to see this shown for e^(-x²) for example.

 

[shmoe]

Try:

Marchisotto and Zakeri, "An Invitation to Integration in Finite Terms", The College Mathematics Journal, Vol 25, No 4 (Sept., 1994) pp 295 - 308

Kasper, "Integration in Finite Terms: the Liouville Theory", Mathematics Magazine, Vol 53, No 4 (Sept 1980) pp 195 - 201.

Conrad's "Imossibility Theorems in the Theory of integration":
http://www.claymath.org/programs/outreach/academy/LectureNotes05/Conrad.pdf

The first two can be found in jstore and give a bit of a general overview. Conrad's goes into details of proofs with the main goal being the Gaussian and logarithmic integrals. There will be oodles more, using "Integration in Finite Terms" will bring up lots of hits, these just happen to be the ones I have on hand. (Alternate English wording- "Integration in Elementary Terms", "Liouville Theory" is another way to go, and of course plunder the bibliographies in the above)

 

[TD]

I never used jstore before, just went to the website and I'll look into that.
Perhaps they have the books at the university library, I'll be checking it soon.

Thanks for the titles and tips, I'll have another try at google as well

 

[shmoe]

I never used jstore before, just went to the website and I'll look into that.
Perhaps they have the books at the university library, I'll be checking it soon.

You'll probably figure it out quickly if you haven't already, but it's jstor, not jstor"e" like I said earlier, the link is http://www.jstor.org/
With an "e" in various forms you get credit card applications, japanamation, sunglasses,...

My internet connection is through my university so I forget sometimes that you need a subscription to access jstor. If you're not in a similar situation you can try at your university library, they'll often have campus wide liscences to stuff like this (MathSciNet and so on). Those two are from pretty common journals though.

Thanks for the titles and tips, I'll have another try at google as well

No problem.