DSE 2013 Question on Intertemporal Choice

[knowpraveen]

Sir, how has the expected demand been derived? Here, the probability distribution of beta is 2 for beta lying in between 1/2 and 1. How did x1=1/1+r come into picture while deriving the expected demand and what does it mean intuitively when beta and x1 are put together in the integral. I couldn't figure it out.

Why didn't we consider the case of beta=1/1+r?