We develop and investigate an integral equation connecting the first passage time distribution of a stochastic process in the presence of an absorbing boundary condition and the corresponding Green's function in the absence of the absorbing boundary. Analytical solutions to the integral equations are obtained for three diffusion processes in time-independent potentials which have been previously investigated by other methods. The integral equation provides an alternative way to analytically solve the three diffusion-controlled reactive processes. In order to help analyze biological rupture experiments, we further investigate the numerical solutions of the integral equation for a diffusion process in a time-dependent potential. Our numerical procedure, based on the exact integral equation, avoids the adiabatic approximation used in previous analytical theories and is useful for fitting the rupture force distribution data from single-molecule pulling experiments or molecular dynamics simulation data, especially at larger pulling speeds, larger cantilever spring constants, and smaller reaction rates. Stochastic simulation results confirm the validity of our numerical procedure. We suggest combining a previous analytical theory with our integral equation approach to analyze the kinetics of force induced rupture of biomacromolecules.