Integral of e^(-x^2): Does it Have an Indefinite Integral?

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SUMMARY

The integral of e^(-x^2) does not have an indefinite integral expressible in terms of elementary functions. However, it can be represented using the error function, erf(x). Specifically, the indefinite integral is defined as ∫ e^(-x^2) dx = (√π/2) erf(x) + C. This conclusion is based on established mathematical principles regarding continuous functions and their integrals.

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Geoffrey
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I know definite integral exists for this between - infinity to + infinity,

does it have an indefinite integral?

Thanks,
Geoff.
 
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Yes. All continuous functions (and, many functions that are not continuous) have indefinite integrals. But the indefinite integral of that function cannot be written in terms of elementary functions. It can, of course, be written in terms of the "error function", erf(x), because that is defined as 2/\sqrt{\pi} times that integral. That is,
\int e^{-x^2}dx= \frac{\sqrt{\pi}}{2}erf(x)+ C
 

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