Integral closure of a subring that is a polynomial ring over an algebraically closed field.

Question

Let $K$ be an algebraically closed field that is a subring of an integral domain $D$. Assume $D$ contains an element $d$ that is transcendental over $K$. Also assume that $D$ is integral over $K[d]$. Must $K[d]$ be all of $D$?

Answer 1

if you pick $D= K[X]$ and $d = X^2$ then $D$ is integral over $K[d]$ but $K[d]$ is not all of $D$.