Background
Recall that a scheme $X$ is called integral at $x$ if $\mathcal{O}_{X,x}$ is an integral domain. This is equivalent to saying that $X$ is reduced at $x$ ($\mathcal{O}_{X,x}$ has no nilpotents) and that there is only one irreducible component of $X$ passing through $x$.
A scheme is called integral if it is reduced (i.e. reduced at all points) and irreducible (i.e. not a non-trivial union of two closed subsets).
It is clear that being integral is not a local property; the disjoint union of two integral schemes is not integral. This suggests the following:
Question
Does there exist a connected scheme that is integral at all point, but not integral? (Connected means that the underlying topological space is connected.)
Note that it is easy to see that such a scheme must have infinitely many irreducible components (for else the irreducible components are open, and local integrality forces them to be disjoint).