Can erf(x) be used to solve e^(x^2)?

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The discussion centers on whether the integral of e^(x^2) can be expressed in terms of elementary functions. While e^(x^2) is a continuous function and has an anti-derivative, it is not an elementary function. The anti-derivative of e^(x^2) cannot be expressed using standard functions like polynomials or exponentials. The Error Function, erf(x), is mentioned as an anti-derivative for e^(-x^2), but it does not directly apply to e^(x^2). Thus, while e^(x^2) has an integral, it does not have a simple expression in terms of elementary functions.
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does the integrate e^(x^2) can solve??
i think is no...
but why??
 
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That depends upon exactly what you mean.

Since e^(x^2) is a continuous function, yes, it HAS an integral (anti-derivative). Every continuous function (and many non-continuous functions) is the derivative of some function and therefore has an anti-derivative.

Is that anti-derivative any "elementary function" (defined as polynomials, rational functions, exponentials, logarithms, trig functions and combinations of them)? No, if fact for most functions the anti-derivative is not an elementary function. (There are more functions in heaven and Earth than are dreamed of in your philosophy, Horatio!)

Of course one can always DEFINE a new function to do the job. I don't know specifically about e^(x^2) but the ERROR FUNCTION, Erf(x) is defined as an anti-derivative of e^(-x^2).
 
eh...

may i ask what is Error Function??
 
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