Let $S$ be the local ring of nodal curve, $R$ = inverse limit $Frob: S \to S$.  For example:

 * $k$ a perfect field,
 * $f(x,y)=y^{p+1}-x^{p+1}(1+x)$,
 * $R=k[x^{1/p^{\infty}},y^{1/p^{\infty}}]/(f^{1/p^{\infty}})$.

Here's a complete local example:

 * $k$ a perfect field,
 * $f(x,y)=y^{p}-x^{p}y-x^{p+1}$,
 * $R=k[[x^{1/p^{\infty}},y^{1/p^{\infty}}]]/(f^{1/p^{\infty}})$.

In each example $(y/x)$ is integral over $R$.