I read the following in a book on differential equations
$$\int \frac{1}{y-1} dy = \log |y - 1|$$
If I put $\int \frac{1}{y-1} dy$ into Wolfram Alpha it gives $\log (y - 1)$, i.e. the argument of the function is not an absolute value unlike the first equation. So which is correct and why am I seeing conflicting results for the same integral?
That happens because $\log (y - 1)$ only exists in $(1, +\infty)$, hence its derivative would be $1/(y - 1)$, but with considering only values of $y$ that are greater than $1$.
This is of course different from what you are trying to integrate, because that $1/(y - 1)$ inside the integral is a function that is defined everywhere in $\mathbb{R}$ (with the sole exception of $1$). Then its indefinite integral is $\log |y - 1|$ because not only its derivative is $1/(y - 1)$, but its domain is also exactly $\mathbb{R} - \{1\}$, that is the same of the function you're trying to integrate.
This happens because functions with different domains are different functions even if their written expression is the same, so $1/(y - 1)$ may yield different results when integrating, according to what its domain is.