# Primitive of exp (-x^2) has no Solution in Elementary Functions

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## Theorem

The primitive:

- $\ds \int \map \exp {-x^2} \rd x$

cannot be expressed in terms of a finite number of elementary functions.

## Proof

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## Historical Note

The proof that $\ds \int \map \exp {-x^2} \rd x$ cannot be expressed with a finite number of elementary functions was first proved by Joseph Liouville.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.29$: Liouville ($\text {1809}$ – $\text {1882}$)