A generalization of the Schwarz integral formula in the class of multiply connected circular domains is constructed. A classical Schwarz integral formula retrieves, up to an imaginary constant, an analytic function in a domain given its real part on the boundary of a domain; Poisson integral formulas are well-known examples for simply connected domains. The generalized integral formulas derived here retrieve an analytic function given more general linear combinations of its real and imaginary parts on each boundary component of a multiply connected domain. Those linear combinations can be different on each boundary component. The chief mathematical tool is the prime function of a multiply connected circular domain. A Schwarz integral formula for such domains, retrieving an analytic function given its real part on all boundary components, was derived in terms of the prime function by Crowdy [Complex Variables and Elliptic Equations, 53(3), 221-236, (2008)]. The present paper combines those formulas with use of radial slit conformal mappings, also expressible in terms of the prime function, to produce integral expressions for analytic functions where more general linear combinations of their real and imaginary parts are specified on the boundary components of a multiply connected domain. We refer to the resulting expressions as generalized Schwarz integral formulas. Their usefulness and versatility are showcased by application to three topical problems: finding the potential around periodic interdigitated electrodes, solving the free boundary problem for hollow vortex wakes behind a bluff body, and determining the two-phase flow over a so-called liquid-infused surface.