I am currently learning stochastic calculus, and while the entire thing is supposedly about defining $\int g(s) dX(s)$ where $X$ is a stochastic process, I don't even understand something like $\int X(s) ds$.
The problem originates in stochastic differentials, where we want solutions to $$X(t) = \int X(s) ds + \int g(s)dW(s)$$ where e.g. $W$ is a Wiener procesc. My book states the above, then proceeds to define the second term ("Ito integral"), and then ends the chapter. But what is the first term?? What is $\int X(s) ds$? How would I calculate such a thing?
X(s) is never given to us concretely, so how could we take its integral?
As mentioned in the comments, $$\int X(s)ds$$ is just a standard integral.
However, this notation suppresses something else that the integral depends on. $X$ is a stochastic process, so it is actually a function of time and some underlying source of uncertainty. That is, $X$ is actually the function $X(t,\omega)$ where $\omega$ lies in some probability space $(\Omega,\mathscr{F},\mathbb{P})$.
Hence, for a fixed $\omega$ (think of this as fixing the realisation of the stochastic process), we can compute the integral as we normally would. The integral is then a random variable defined on the same probability space $X$ is.