Let $k$ be a field of characteristic $2$. Let $f(x) \in k[x]$ be a polynomial of positive degree without double roots in $k$.
Also assume that $k$ is algebraically closed and perfect.
Then $f'(x)$ is a square in $k[x]$.
Write it $f'(x):=(\prod_{i=1}^{t}(x-b_i)^{r_i})^2.$
I'd like to show that
The elements $$\alpha_i := \frac{\sqrt{f(b_i)}-\sqrt{f(x)}}{(x-b_i)^{r_i}}, \quad i=1, \dots, t,$$ are integral over $k[x]$
And also
The set $\{1, \alpha_1, \dots, \alpha_t\}$ generates the integral closure $B$ over $k[x]$ in $k(x)(\sqrt{f})$.