Say:
$f'(x) = e^{-x^2}.$
Then, what is $f(x)$ ?
I ask this because I heard:
$$\int_{-\infty}^{\infty} e^{-x^2}\, dx=\sqrt{\pi}.$$
Thus curious about $f(x),$ but can't figure out it on my own, though tried.
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Thomas Andrews
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Eric
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2We can not get it in the elementary functions. – Michael Rozenberg Mar 13 '20 at 02:33
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3Does this answer your question? Is there really no way to integrate $e^{-x^2}$? – Matheus Nunes Mar 13 '20 at 02:37
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2It is rigorously proved that the antiderivative of $e^{-x^2}$ cannot be expressed in elementary terms. The situation is not so different from that of the square root function where we could not write it as a finite combination of arithmetic operations and had to add it as a new vocabulary to the dictionary of functions. – Sangchul Lee Mar 13 '20 at 02:43