They share a Latin root corresponding the the concept of 'wholeness'. In the context of integers, this would be interpreted as 'whole numbers' - i.e. numbers with no fractional part. In the context of integration, this would correspond to 'summing up to create a whole' in the sense of the integral representing a continuous sum or area. I think that's why they use similar words.
I'm no expert in such things, but I've included some links to an online etymology dictionary that appears to agree with me.
1: Integer: From the Latin adjective integer, "whole, complete"
2: Integrate: "to render something whole"/"to put together parts or elements and combine them into a whole"
3: Integral: "of or pertaining to a whole"