$R[x]$ can be an integral extension of $R$?

Question

Is the polynomial ring $R[x]$ an integral extension of $R$, if $R$ is a domain?

Answer 1

In general, the polynomial ring $R[x]$ is not an integral extension of $R$: Note that $x^n + \sum_{i=0}^{n-1} a_i x^i \neq 0$ for all $n, a_i$ (for a formal proof, take the degrees of both sides).

In fact, $R[x]$ is one of the go-to examples for a transcendental extension.

Answer 2

Another possibility, besides Johannes Kloos's approach is to recall the following:

A ring extension $S/R$ is integral, if and only if for every $\alpha\in S$ one has that $R[\alpha]$ is an f.g. $R$-module. Does this hold true when $S=R[x]$?