How can one prove that integral can not be solved in transcendental functions?

Question

For people who do not have time to read all this:

Is there a simple proof (restricted to knowing calculus) that the integral can not be solved in transcendental function.

Longer version:

I am curious how people were able to understand that some particular integral (for example $\int e^{x^2}$) can not be solved in transcendental functions?

In my opinion it is so counter intuitive, and after knowing that almost every function can be differentiated (I am aware that not all), and that for every integral there exist a function that should be anti-derivative I would assume that the integral of everything should be a transcendental function.

All in all, if people where trying for long time to find $\int e^{x^2}$ and was not able to find a transcendental function, it does not mean that it does not exist. I am glad that math was not done by me, but how have they proved that some integrals can not be taken?

The only thing I know up till now is the Liouville theorem, http://www.sci.ccny.cuny.edu/~ksda/PostedPapers/liouv06.pdf. But this is a generalization.

can you identify any function by a sum of truncated Taylor series? — Seyhmus Güngören, Mar 14 '13 at 11:11
duplicate: http://math.stackexchange.com/questions/265780/ http://math.stackexchange.com/questions/155 — sdcvvc, Mar 14 '13 at 11:12
@sdcvvc thanks, the first answer is actually a really good answer, but, rather a complicated one. About the second - this is what I have mentioned in the part about Liouville theorem. — Salvador Dali, Mar 14 '13 at 11:28
There is a well-developed theory of "integration in finite terms", associated with Ritt. Have a look on the web. — Gerry Myerson, Mar 14 '13 at 11:39

How can one prove that integral can not be solved in transcendental functions?

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