"Integration" suggests a set of cultural attitudes and a culture that can be integrated into.
Although it may well rely on that, in practise it means the level of cohesion in civil society including all of the relations, institutions that exist with in it.
Self alienation and self atomisation is one of the causes of whatever breakdown in social cohesion exists in a society, and in my opinion, is perhaps the primary cause in our particular context. So to say that culture is not fundamental to the question is ridiculous.
'Our particular context' = Britain? Or muslims in Britain?
The implication of the OP is pretty clear that cultural differences stand in the way of 'harmony' (maybe dogs no longer chasing cats, and the planets aligning too). This is the typical narrative against multiculturalism. Yet both stem from similar grounds.