Indeed, you are right: in problem 1 the width of the well is 2a, not a as I listed originally. The well extends from x=-a to x=a, and has V=0 over that interval. For |x|>a, the potential is infinite. That is why the normalization constant is 1/sqrt(a), not sqrt(2/a).
I've corrected that on the problem set, available at the same place.