Plugging into mathematica with specific values for b you get

b = 1.1, integral = 8.04619

b = 2, integral = 0.989315

b = 10, integral = 0.101846

b = 100, integral = 0.00999388

This seems close to 1/(b-1) and I wouldn't be too suprised if the discrepancies are due to the fact this integral is poorly behaved.