How can something be proved unsolvable? My question specifically deals with certain real indefinite integrals such as $$\int e^{-x^2} {dx} \ \ \text{and} \ \ \int \sqrt{1+x^3} {dx}$$ Books and articles online have only ever said that these cannot be expressed in terms of elementary functions. I was wondering how this could be proved? I know this is a naive way of thinking, but it seems to me like these are just unsolved problems, not unsolvable ones.

No, these are theorems.  See the Risch algorithm.   The basic idea is that it is possible to show that if there is an elementary function with the given integrand as derivative, it must have a certain general form.
If nothing of that form works, there is no elementary antiderivative.
