f(x) = abax-(a+1) or Cx-(a+1) where C = aba (a constant)
Remember that the pdf is defined over the domain [b, inf] otherwise zero.
Mean = integral xf(x) evaluated from b to infinity.
Remember also that the limit of 1/x as x goes to infinity = 0. Similarly for any positive a, (1/x)a goes to 0 as x goes to infinity.
mean = integral C x-(a+1)x dx = integral Cx-a = C(1/(-a+1))x-a+1 evaluated over the interval b to infinity.
The integral is zero at infinity, so the mean = C(0-1/(-a+1))b-a+1
Remember b-a+1 = b-ab
After substituting and cancelling
mean = ab/(a-1) for a greater than 1.