Is there any idea to prove that $F + xk[[x]]$ is not integrally closed when the field $k$ is a proper extension of the field $F.$
I claim the integral closure of the ring $R:=F+xk[[x]]$ is $R':=F'+xk[[x]]$, where $F'$ is the algebraic closure of $F$ inside $k$. For one, the field of fractions of $R$ is $k((x))$, and $F'$ is integral over $R$, so the integral closure of $R$ is at least as large as $R'$. On the other hand, $R$ is contained in power series ring $k[[x]]$, which is integrally closed, so any element $f$ of the integral closure of $R$ had better be a power series in $x$. By evaluating its minimal polynomial at $x=0$, we see that the constant term of $f$ is integral over $F$, so $f\in R'$, as desired.