Please explain the steps in the following solution given (a question of integration using substitution)?

They are trying to write (x-1) as a differential function of the denominator and matching the coefficients A and B such that they get a simple du/u to integrate. They then split up the differential function into the parts A and B and integrate them separately. The part with the B term is a bit more difficult because it involves trig substitutions, but at the end they add the A and B parts back together to give the overall answer.