The Wallis Formula for Pi and the Dawson Function

I spent quite some time watching this video on the derivation of the Wallis product and practised until I could reproduce it without any external assistance. A crucial part of the solution relies on integration by parts to set up a reduction formula for the integral of \( sin(x)^n\). Here is the very well-presented and easy to understand video:

Note: the following discussion centres on integration by parts and is not related to the Wallis function.

I sometimes practise using integration by parts to solve integrals that I think of and last night my mind fell on a deceptively easy-looking integral, namely \(e^{x^2}\). The graph of this function is well-behaved and I thought that there would be an easy solution but try as I might I couldn't find it. Reluctantly, I checked first Symbolab and then WolframAlpha to find out how it could be done. Here's what the former had to say:

What on earth is this F(x) that just appears out of nowhere? WolframAlpha offered the same solution but had accompanying documentation that described F(x) as the Dawson integral defined by:

There is a quite comprehensive article about the Dawson integral or Dawson Function, as it's alternatively called, on Wikipedia but it's largely incomprehensible to me at the moment. Maybe I can come to terms with it later.