Proving $R[Y_1, \ldots, Y_r]_{P}$ is integrally closed (trying to prove $\mathbb{P}^r_R$ is normal)

Question

Let $R$ be an integral domain with algebraically closed fraction field. Let $P$ be a prime ideal of $R[Y_1, \ldots, Y_r]$. Then it follows that $R[Y_1, \ldots, Y_r]_{P}$ is an integral domain. I would like to prove that $R[Y_1, \ldots, Y_r]_{P}$ is integrally closed in its fraction field. Any explanation would be appreciated. Thank you.

Edit. I am asking this question because I wanted to prove that $\mathbb{P}^r_R$ is a normal scheme when $R$ is a valuation ring with an algebraically closed fraction field.

Answer 1

This is a community wiki answer recording the discussion from the comments, in order that this question might be marked as answered (once this post is upvoted or accepted).

If $R$ is a valuation ring then it is integrally closed, then so is any polynomial ring over $R$, and its localizations as well. – user26857