The injective hull of a polynomial ring in one variable $K[X]$ (where $K$ is a field) is $K(X)$ since $K(X)$ is a divisible hence injective $K[X]$-module (since $K[X]$ is a PID) and $K(X)$ is an essential extension of $K[X]$.
What about more than one variable?
Claim: If $A$ is an integral domain with field of fractions $Q(A)$, then $_AA\hookrightarrow Q(A)$ is an injective hull.
Proof. Firstly, $Q(A)$ is injective as an $A$-module since $$\text{Hom}_A(-,Q(A))\cong\text{Hom}_{Q(A)}(-\otimes_A Q(A),Q(A)):A\text{-Mod}^{\text{op}}\to{\mathbb Z}\text{-Mod}$$ is exact as the composition of the exact localization functor $$-\otimes_A Q(A):A\text{-Mod}\to Q(A)\text{-Vect},$$ the exact $Q(A)$-dualization functor $$\text{Hom}_{Q(A)}(-,Q(A)): Q(A)\text{-Vect}^{\text{op}}\to Q(A)\text{-Vect}$$ and the exact forgetful functor $Q(A)\text{-Vect}\to{\mathbb Z}\text{-Mod}$.
Further, the natural map $A\to Q(A)$ is an essential extension, so the statement follows.