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Specifically, why is there no antiderivative, or any possible method of integrating (except numerically) say $\;e^{\csc(x)}$?

(I don't have my computer handy right now so I cant format the formula, sorry about that!)

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It's not that it doesn't have an antiderivative but that there's no expression for it in terms of a finite number of elementary functions. –  Sharkos Jun 5 '13 at 0:07
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It's kind of similar to as saying: Why can we foil two linear terms to become a quadratic, but not any given quadratic can be factored into (real) linear terms (anti Foil, if you wish). That's just how it is. The far majority of continuous functions do not have an anti derivative in terms of elementary functions (that's I think what you mean here), and thus we need to resort to numerical methods to find area under the curve, or whatever the integral stands for.

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Yeah that's what I meant. I guess the follow up is, can we easily figure out when a function has no antiderivative? In your case of antifoiling it's easy to figure out, but say with the Gaussian integral, I honestly wouldn't have known that it was integrable with Gauss' good help. –  Soyuz Jun 5 '13 at 0:10
    
It wasn't Gauss, it was Liouville. –  André Nicolas Jun 5 '13 at 0:17
    
Can you explain? I hate to be misinformed on the history. And on a side note Risch's algorithm would be solution , though I was hoping for something simpler... –  Soyuz Jun 5 '13 at 0:21
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As far as the anti foiling is concerned, it is easy to figure out whether or not a quadratic expression can be factored. The discriminant can tell you that. With higher degree polynomials, it isn't always that easy to see if it is factorable. With integrals it is similar in a way, except the process of anti deriving is more complicated. There is no discriminant or such thing that determines whether or not a function has an anti derivative. Even today I am sure there are functions that do have an anti derivative in elem. functions, but they are so big, that with computers we don't find them. –  imranfat Jun 5 '13 at 0:21
    
Forgive my ignorance, but was it Galois or someone who proved that there are no closed form for some integrals? –  sidht Jun 5 '13 at 1:45
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