how to integrate a/1-ay?

In private correspondence with the author of the question, it was determined

that 'y' is the variable of integration.

Therefore, 'a' is a constant for purposes of this integration, and that whole

'a' business can be taken outside the integral.

Integral [a / (1-ay) dy] = a / (1-a) integral [dy/y] = a ln (y) / (1-a) + C