  We present the integrated 3-point shear correlation function $i\zeta_{\pm}$
-- a higher-order statistic of the cosmic shear field -- which can be directly
estimated in wide-area weak lensing surveys without measuring the full 3-point
shear correlation function, making this a practical and complementary tool to
2-point statistics for weak lensing cosmology. We define it as the 1-point
aperture mass statistic $M_{\mathrm{ap}}$ measured at different locations on
the shear field correlated with the corresponding local 2-point shear
correlation function $\xi_{\pm}$. Building upon existing work on the integrated
bispectrum of the weak lensing convergence field, we present a theoretical
framework for computing the integrated 3-point function in real space for any
projected field within the flat-sky approximation and apply it to cosmic shear.
Using analytical formulae for the non-linear matter power spectrum and
bispectrum, we model $i\zeta_{\pm}$ and validate it on N-body simulations
within the uncertainties expected from the sixth year cosmic shear data of the
Dark Energy Survey. We also explore the Fisher information content of
$i\zeta_{\pm}$ and perform a joint analysis with $\xi_{\pm}$ for two
tomographic source redshift bins with realistic shape-noise to analyse its
power in constraining cosmological parameters. We find that the joint analysis
of $\xi_{\pm}$ and $i\zeta_{\pm}$ has the potential to considerably improve
parameter constraints from $\xi_{\pm}$ alone, and can be particularly useful in
improving the figure of merit of the dynamical dark energy equation of state
parameters from cosmic shear data.
