To find the integral of the function $\int \frac{1}{x + \sqrt{x - 1}} \, dx$, let's proceed with the following steps:

Step 1: Substitution

Let us use the substitution: $u = \sqrt{x - 1}$ This implies: $u^2 = x - 1 \quad \Rightarrow \quad x = u^2 + 1$ Differentiating both sides with respect to $x$: [ \frac{du}{dx} = \frac{1}{2\sqrt