  We present the integrated 3-point correlation functions (3PCF) involving both
the cosmic shear and the galaxy density fields. These are a set of higher-order
statistics that describe the modulation of local 2-point correlation functions
(2PCF) by large-scale features in the fields, and which are easy to measure
from galaxy imaging surveys. Based on previous works on the shear-only
integrated 3PCF, we develop the theoretical framework for modelling 5 new
statistics involving the galaxy field and its cross-correlations with cosmic
shear. Using realistic galaxy and cosmic shear mocks from simulations, we
determine the regime of validity of our models based on leading-order standard
perturbation theory with an MCMC analysis that recovers unbiased constraints of
the amplitude of fluctuations parameter $A_s$ and the linear and quadratic
galaxy bias parameters $b_1$ and $b_2$. Using Fisher matrix forecasts for a
DES-Y3-like survey, relative to baseline analyses with conventional
3$\times$2PCFs, we find that the addition of the shear-only integrated 3PCF can
improve cosmological parameter constraints by $20-40\%$. The subsequent
addition of the new statistics introduced in this paper can lead to further
improvements of $10-20\%$, even when utilizing only conservatively large scales
where the tree-level models are valid. Our results motivate future work on the
galaxy and shear integrated 3PCFs, which offer a practical way to extend
standard analyses based on 3$\times$2PCFs to systematically probe the
non-Gaussian information content of cosmic density fields.
