  We study perturbative aspects of recently proposed integrated four-point
correlators in $\mathcal{N}=4$ supersymmetric Yang-Mills with all classical
gauge groups using standard Feynman diagram computations. We argue that
perturbative contributions of the integrated correlators are given by linear
combinations of periods of certain conformal Feynman graphs, which were
originally introduced for the construction of perturbative loop integrands of
the un-integrated correlator. This observation allows us to evaluate the
integrated correlators to high loop orders. We explicitly compute one of the
integrated correlators up to four loops in the planar limit, and up to three
loops for the other integrated correlator, and find agreement with the results
obtained from supersymmetric localisation. The identification between the
integrated correlators and certain periods also implies non-trivial relations
among these periods, given that one may compute the integrated correlators
using localisation. We illustrate this idea by considering one of the
integrated correlators at five loops in the planar limit, where the
localisation result leads to a prediction for the period of a certain six-loop
integral.
