Is this modified Dirichlet function integrable and/or uniformly integrable?

Question

I was considering the following functions $X_n(x)$ , similar to the Dirichlet function:

\begin{Bmatrix} q\: \; \textrm{when} \; x\:=\frac{p}{q} \in \mathbb{Q} \cap [0, \frac{1}{n}]& \; p \in \mathbb{N}\;\textrm{and} \; q \in \mathbb{N} \; \textrm{in lowest form}\\ 0 \; \textrm{otherwise} & \end{Bmatrix}

1. Are the Xn integrable ? [I think 'yes' and the integrals are 0 because the rationals are countable, although I am not sure if the unboundedness messes things up ...]
2. and if so, are they uniformly integrable ? [I think 'yes' because the set on which it is unbounded is of measure zero but I am not sure ...]