I don't understand how to define such integration factor in order to solve SDE, for example, as was shown in Solving $dX_{t} = \mu X_{t} dt + \sigma dW_{t}$ and Solving Stochastic Differential Equation using integrating factor. And in this book Stochastic Differential Equations in some exercises hints recommend to use integrator factor, but how to get it.
Is there some mechanism(algorithm) which allows us to find this integrator factor?
I believe that Wikipedia gives a fair definition ; see it as a "helper function".
In your example, $e^{-\mu t}$ helps you getting rid of the drift in your SDE, and you end up solving it by mere stochastic integration ; then just "multiply back" by $e^{\mu t}$ and your initial SDE is solved.
Essentially you want to get rid of the drift of the process $\mu X_t d t$ to get it in a solvable form
$$E[d X_t] = E[\mu X_t d t]\\ \implies d E[X_t] = \mu E[X_t] d t\\ \equiv d y(t) = \mu \cdot y(t) d t\\ \implies \frac{d y(t)}{d t} - \mu \cdot y(t)=0$$
Hence solving this ODE via the integrating factor method one has
$$I(t) = e^{\int_0^t -\mu d t} = e^{-\mu t}$$
So getting the SDE
$$d (e^{-\mu t} X_t) = X_t \cdot d(e^{-\mu t}) + e^{- \mu t} \cdot d X_t + (d e^{- \mu t})(d X_t)\\ = X_t \cdot (- \mu e^{- \mu t}dt) + e^{- \mu t}(\mu X_t d t + \sigma d W_t) + 0\\ = - \mu e^{- \mu t}X_t d t + \mu e^{-\mu t}X_t d t + \sigma e^{- \mu t }d W_t\\ = \sigma e^{-\mu t} d W_t$$
Hence we have removed the drift and get $$e^{-\mu t} X_t = X_0 + \sigma \int_0^t e^{-\mu t} d W_t$$
From there it should be obvious. This method should work with any SDE with drift term $\mu(t)X_t d t$
