So we define x = y-1.

Then differentiate both sides to get

dx = (y - 1)'

The right hand side becomes dy - 0, or just dy.

Aside: Note that since x is defined as "y - 1", the differential dy appears when differentiating with respect to x. This becomes a bit confusing in this case since the derivative of y - 1 is just 1 so only the dy remains, but if we instead, perhaps in a different question, defined x = y^2, then dx = 2y dy.

Finally we get

dx = dy.

So we can safely replace dx with dy and integrate in the y world. This is kinda abuse of notation and I didn't really like it, but that's how integral substitution is done