Am I right to think that though we have a well-established theory of integrability of functions by means closed-form expressions, we have nothing similar about definite integrals ?
For instance, the Gaussian is well-known to have no closed-form antiderivative (this is proven), but the integral from $0$ to $\infty$ is $\sqrt\pi$.
Update:
When I say closed-form for a constant, I mean a closed-form expression where only integer arguments are allowed. For example, $\pi^e$ as it equals $\arccos^{\exp(1)}(-1)$, but not $\gamma$.
If you really want some kind of information that takes a definite integral (with arbitrary integrands? or restricted to some manageable class?) (over arbitrary domains? or restricted?), and determines whether there is closed-form expression, then I suspect it is rather hopeless. Or at least impressively challenging.
However you might be interested in the theory of periods, introduced by Kontsevich and Zagier. This is, as you ask for, a "well-established theory", but still highly active, with major conjectures, open questions, etc. In a nutshell, and slightly oversimplifying, a period is a number that is equal to the definite integral of a rational function (quotient of polynomials) with rational ($\mathbb{Q}$) coefficients, over a domain defined by polynomial inequalities with rational coefficients.
Every algebraic number is a period. $\pi$ is a period: it is the definite integral of $1$ over the domain $x^2+y^2 \leq 1$ (a polynomial inequality with rational, indeed integer, coefficients). One of the open questions is whether $e$ and $1/\pi$ are periods.
I only mentioned here the issue of deciding whether a given number is a period (i.e., equal to a definite integral of a certain form). This is by no means the only theme; e.g., structure of the set of periods, connections to algebraic geometry, etc.
See: https://arxiv.org/pdf/1407.2388.pdf .