# How to solve $\int\limits_0^p e^{x^2} dx$?

Integration by parts does not seem to work. I was wondering if this integral could be solved using a specific contour and applying for example Jordan's lemma?

-
This is the imaginary error function. – user26872 Mar 21 '12 at 2:15
"Find" or "evaluate" is the right word here; "solve" is not. One solves equations; one solves problems. One evaluates expressions. In this case, one seeks a value, not a solution. – Michael Hardy Mar 21 '12 at 2:17
I think this is what @MichaelHardy is trying to say... – The Chaz 2.0 Mar 21 '12 at 2:19
Hmm I remember you have to square this integral and use spherical coordinates. – FiniteA Mar 21 '12 at 2:34
One can solve the problem of evaluating the integral, though :) – Mariano Suárez-Alvarez Mar 21 '12 at 2:54

Elementary functions are generally considered to be functions generated by the identity function, constant functions, basic trig functions, inverse trig functions, exponential functions, logarithmic functions, the four arithmetic operations, and composition. It is well-known that while $\mathrm{e}^{x^2}$ has an antiderivative, that antiderivative is not among the elementary functions.
A name has been given to a similar function $F$, where $F(p)=\frac{2}{\sqrt{\pi}}\int_0^{p}\mathrm{e}^{-x^2}\,dx$. The standard name of this function is the error function, denoted $\operatorname{erf}$. Note the minus sign in the exponent. This explains the name that oenamen gives to your function in his comment.