Corollary

Let $R$ be a ring and $A, B, C$ subrings $\& R$ such that $A \subset B$. If $B$ is integral over $A$, $C[B]$ is integral over $C[A]$.

$B \otimes_A C[A]$ is integral over $C[A]$ by Proposition 5 and hence so is the canonical image $C[B]$ of $B \otimes_A C[A]$ in $R$ (considered as an A-algebra) by Proposition 2.