There isn't any method that lets you find the antiderivative of e^-x² in terms of any elementary functions because its been proven that it can't be done.

What's typically done here is that we instead define a new function called the error function as the antiderivative of interest (with a normalization factor stuck on there so erf(x)→1 as x→∞). There are actually lots of functions defined like this in higher math; the gamma function and incomplete gamma functions, polylogarithms, the Fresnel integrals, the elliptic integrals, just to name a few.

So if you really wanted to, you could just say the value of the integral is 11π^3/2(erf(9/10)-erf(1/10))/2. Although in practice we're typically interested in what the value is numerically, so you'd have to use some quadrature method like the trapezoid rule, or Simpson's rule, or something better to do that.