# How do you know when an expression is not solvable?

My question pretty much is the title of this post, but let me explain it a bit more. When in physics or mathematics, or some other discipline involving math, you run into an integral, a differential equation or some other expression, how do you know if it's solvable or not? I mean, how do I know when to tend to numerical computations because an analytical result cannot be obtained? How do I know that my incapability to solve a certain equation is not just due to my lack of knowledge, but because the equation is, in fact, now solvable?

For example, how would I know that the integral:

$$\int e^{-x^2} dx$$

is not solvable?

I don't think there is anyway of knowing whether an integral is possible or not unless it's been proven one way or another for that specific integral.

morphism