Role of AdS/CFT correspondence in the context of integrability

I would recommend the review

N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99, 3 (2012), arXiv:1012.3982.

For example, here is an excerpt from p.10:

Methods of integrability provide us with reliable data over the complete range of couplings. We can investigate in practice a gauge theory at strong coupling. There it behaves like a weakly coupled string theory. Likewise a string theory on a highly curved background (equivalent to low tension) behaves like a weakly coupled gauge theory. At intermediate coupling, the results are reminiscent of neither model or of both; this is merely a matter of taste and depends crucially on whether one’s intuition is based on classical or quantum physics. In any case, integrability can give us valuable insights into a truly quantum gauge and/or string theory at intermediate coupling strength.

and p.11:

A crucial benefit of integrability is that the spectral equations include the coupling constant λ in functional form. Whereas standard methods produce an expansion whose higher loop coefficients are exponentially or even factorially hard to compute, here we can directly work at intermediate coupling strength.

Having a weakly coupled gravity dual does not imply integrality. To the contrary black holes in classical gravity are the most chaotic objects there can be! See this article:

https://arxiv.org/abs/1503.01409