# Evaluation of an integral

## Main Question or Discussion Point

Does anyone know how to go about solving

$\int e^{(x^{2})} dx$

not expressible in terms of elementary functions. Using the "special" function, $\mbox{erf} (x)$ (the "error function", defined by $\mbox{erf}(x) = 2/\sqrt{\pi} \int_0^x e^{-t^2} \ dt$), you can express the integral in this way, though:

$$\int e^{(x^2)} \ dx = -\frac{i\sqrt{\pi}}{2}\mbox{erf}(ix)+C.$$

Essentially what this means is that you either have to compute definite integrals with this integrand numerically, or look up values on tables for the error function.

Data said:
not expressible in terms of elementary functions. Using the "special" function, $\mbox{erf} (x)$ (the "error function", defined by $\mbox{erf}(x) = 2/\sqrt{\pi} \int_0^x e^{-t^2} \ dt$), you can express the integral in this way, though:

$$\int e^{(x^2)} \ dx = -\frac{i\sqrt{\pi}}{2}\mbox{erf}(ix)+C.$$

Essentially what this means is that you either have to compute definite integrals with this integrand numerically, or look up values on tables for the error function.
This is helpful. I have the bounds. Thaks. It is weird that there is no analytical solution to something that looks so simple. I did write a taylor series and integrated that to get an infinite sum that is equal to the integral. It is:

$\sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)n!}}$

that's like the classic example of something that has no elementary solution...

dextercioby
Homework Helper
The "erf" function is tabulated for real arguments only...

Daniel.

dextercioby