A stick of length 1 is split at a point U that is uniformly distributed over (0,1). Determine the expected length of the piece that contains the point p, 0 < p < 1. What is this expected length when p = 1/2?

Can you please explain me how to simplify the integral? From the book i know the answer will be 1/2+p(1-p) before we plug in the value for p.