Difference of order statistics in a sample of uniform random variables

One does not need to evaluate the integral. The change of variables $u=(1-w)v$ yields that $f_W(w)$ is a constant factor $C$ times $w^{s-1-r}$ times $(1-w)^{n+r-s}$. This also yields the value of the constant factor $C$ since $f_W$ must integrate to $1$.

(Or, after the change of variables, one can recognize the integral over $v$ from $0$ to $1$ as a Beta.)