Finding an integral subring of $k[x_1, ..., x_n]/P$ generated by $n$ algebraically independent elements

Question

Let $R = k[x_1, ..., x_n]/P$, where $x_i$'s are variables and $P$ is prime. Suppose $R$ has transcendence degree $n$. I would like to find $z_1, ..., z_n \in R$ algebraically independent such that $R$ is integral over $k[z_1,..., z_n]$. This is the base case in the induction proof of Noether's normalization lemma in the book I am looking at. But it skips the details. Any explanation is appreciated!