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I already know that $\int_{-\infty}^{\infty} e^{-x^2}dx = \sqrt{\pi}$ and by the fundamental theorem of calculus this could be computed by its antiderivative function but it`s impossible to find.

I ask my teachers for proofs for this but they always tell me to investigate in some obscure book. I also tried to look for an idea of the proof (what techniques are used or a sketch of it) but I can`t find any.

Gauss 'n Roses
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    This is because it is a genuinely difficult result, both to precisely state and to prove. – Aphelli Apr 09 '22 at 18:18
  • It is less that it is impossible to find and more that it is impossible to express in a "nice" way. E.g. I can tell you that one antiderivative is: $$\int_0^x e^{-t^2},\mathrm{d}t$$Or that one antiderivative is: $$\sum_{n=0}^\infty\frac{(-1)^n}{n!\cdot(2n+1)}x^{2n+1}$$It's just that neither is analytically very useful. There is something called the "Risch algorithm" I believe, that will show if it is "nicely expressible". Mindlack's comment sums the issue up – FShrike Apr 09 '22 at 18:23
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    The field that deals with this is called differential Galois theory. You may have a look at this: https://www.math.purdue.edu/~adebray/lecture_notes/u19_differential_Galois_theory.pdf – Jean-Claude Arbaut Apr 09 '22 at 18:23
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    Well, they're telling you to look at obscure books because you're really about to open Pandora's box here. The keyword is "Differential Galois Theory". I know people with PhD's who have never heard of this. I'm absolutely no expert in this, but the first reference that comes to mind is Andy R. Magid, Lectures on Differential Galois Theory, University Lecture Series Volume: 7; AMS (1994) – Ivo Terek Apr 09 '22 at 18:23

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