first obtain the density of wi=-ln yi
then the sum of iid copies of this should be easy
you need not integrate at all if you realize what the distribution is.
Let Y1,...,Yn be a random sample from pdf:
f(y|θ)= θy^(θ-1) , 0<y<1, θ>0
and 0 elsewhere
Given that ∑ -ln(Yi) is sufficient for θ,
a) What is the MVUE for θ?
b) Show that E(1/(2θ∑Wi)) = 1/(2(n-1)) , where Wi = -ln(Yi)