In pattern recognition, data integration is a processing method to combine multiple sources so that the combined result can be more accurate than a single source. Evidence theory is one of the methods that have been successfully applied to the data integration task. Since Dempster-Shafer theory as the first evidence theory can be against our intuitive reasoning with some data sets, many researchers have proposed different rules for evidence theory. Among all these rules, the averaging rule is known to be better than others. On the other hand, α-integration was proposed by Amari as a principled way of blending multiple positive measures. It is a generalized averaging algorithm including arithmetic, geometric and harmonic means as its special case. In this paper, we generalize evidence theory with αintegration. Our experimental results show how our proposed methods work.