The elements of a regular sequence in $k[x_1, \ldots, x_n]$ are algebraically independent over $k$ (see for example Matsumura ex. 16.6), and so for a length n regular sequence $(f_i)$, $k(x_1, \ldots, x_n)$ will be algebraic over $k(f_1, \ldots, f_n)$.

My question is whether it's also true that $k[x_1, \ldots, x_n]$ is integral over $k[f_1, \ldots, f_n]$.  Aside from experimental evidence (dozens of randomly generated regular sequences in Macaulay with n=3 and low-ish degrees), I don't really have any reason to think that it should be true, and I haven't been able to prove or disprove it (or find a mention of it anywhere).

Thanks.