I want to prove the remark 3.2.9 of the book Algebraic Geometry of Arithmetic Curves (of Quing Liu) that is: let $X$ be an integral scheme with function field $K(X)$, if $K(X)\otimes_k \overline{k}$ is integral then $X$ is geometrically integral (that is $X_\overline{k}$ is an integral sheme). The proof is going so: for all open $U\subseteq X$, one has $\mathcal{O}(U_\overline{k})\simeq\mathcal{O}(U)\otimes_k \overline{k}\subseteq K(X)\otimes_k \overline{k}$. So $\mathcal{O}(U_\overline{k})$ is integral, forall $U$ and so $X_\overline{k}$ is integral.
My problem is the last implication: I know that a scheme $Y$ is integral if for all open $V\subseteq Y$, $\mathcal{O}(V)$ is integral. But in the proof the open $U_\overline{k}$ are not all the open (I guess). So how to conclude.
Maybe the $U_\overline{k}$ are all the open?
Maybe there is a criterion for integral scheme that would be: $X$ is integral if $X$ is cover by $U_i$ such that for all $i$, $\mathcal{O}(U_i)$ is integral?
The following is true: if $X$ is an irreducible scheme such that there exists an open cover by affine integral schemes, then $X$ is integral. In your case, you can get an open cover with affin open sets of the form $U_{\bar{k}}$.