Does the equation $\int e^{-x^2}dx =1$ have a solution in the set of real numbers? It seems that it doesn't because if I take the derivative of both sides I get $e^{-x^2}=0$ which obviously has no solutions except if we admit infinity as a solution. I do not know much about non-elementary functions so I am not sure that my answer is correct, I am not even sure if I am asking the right question. (-:
2026-03-26 14:19:04.1774534744
Values of non-elementary functions
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The indefinite integral $\int e^{-x^2}\;dx$ is a function, not a number. It is not a constant function, in particular not the constant $1$.
However, there is a solution $a$ for $$ \int_{-\infty}^a e^{-x^2}\;dx = 1 $$ Approximately $0.114$.
Now the equation is a number equal to a number.
Similar question: Is there a number $x$ so that $\int x^2 \;dx = 1$? Same objection. Now $\int x^2\;dx = \frac{x^3}{3}$, and there is, indeed, a number x so that $\frac{x^3}{3}= 1$. Namely $x=3^{1/3}$ has that property. But it is not legitimate to "differentiate" $\frac{x^3}{3}= 1$ and get $x^2 = 0$.