Using right derived functors of the global sections functor, I'd like to calculate the first cohomology group of the constant sheaf $\mathbf{Z}$ on $S^1$ with its usual topology, $H^1(S^1,\mathbf{Z})$. I understand this can be done using Čech cohomology, but I'd like to compute it with derived functors.
To do so, I need to write down an injective resolution of the constant sheaf $\mathbf{Z}$. One such injective resolution involves products of skyscraper sheaves, but this seems fairly messy if all I want to show is $H^1(S^1,\mathbf{Z})=\mathbb{Z}$. Is there a cleaner injective resolution of $\mathbf{Z}$ in $\mathfrak{Ab}(X)$, the category of sheaves of abelian groups on $X$?