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The classic book Calculus made easy includes the following statement:

No one, even today, is able to find the general integral of the expression,

$\frac{dy}{dx} = a^{-x^2}$,

because $a^{-x^2}$ has never yet been found to result from differentiating anything else.

Is anyone aware of a proof for this statement?

Hagbard
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    Your text phrased this poorly; it is known that this can not be integrated in elementary functions. See this paper for example. – lulu Dec 17 '21 at 14:08
  • Thank you very much for your comment. I rephrased the question. – Hagbard Dec 17 '21 at 14:18
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    I don't see any significant changes. The paper I linked to should give you a strong introduction to the proof. As you might imagine, the first hard stage is to define precisely what one means by "integrated in elementary functions". Intuitively, this is clear enough but getting the details straight is critical. The paper goes into all that. – lulu Dec 17 '21 at 14:21
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    The book being from 1910 explains why the author is unaware that the reason nobody had done it is that nobody can do it. – Thomas Andrews Dec 17 '21 at 14:28
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    @ThomasAndrews It seems this fact was proved by Liouville in 1835. Maybe I don't quite understand your comment. – podiki Dec 17 '21 at 15:07

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A much better phrasing of the idea that your textbook is trying to convey is that $a^{-x^2}$ is not the derivative of any sum, multiplication, or composition of rational functions over the complex numbers, radical functions over the real numbers, trigonometric functions over the real numbers, logarithmic functions over the real numbers, or inverses thereof. Also, it is not that no such function with $a^{-x^2}$ as its derivatives is know. Rather, it is more that it has been proven that no such function can have $a^{-x^2}$ as its derivatives.

Angel
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