When integrating data coming from multiple different sources we are faced with the possibility of inconsistency in databases. In this paper, we use one of the paraconsistent logics introduced in [9,7] (LFI1) as a logical framework to model possibly inconsistent database instances obtained by integrating different sources.We propose a method based on the sound and complete tableau proof system of LFI1 to treat both the integration process and the evolution of the integrated database submitted to users updates. In order to treat the integrated database evolution, we introduce a kind of generalized database context, the evolutionary databases, which are databases having the capability of storing and manipulating inconsistent information and, at the same time, allowing integrity constraints to change in time. We argue that our approach is sufficiently general and can be applied in most circumstances where inconsistency may arise in databases.