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Quantifying signal transmission in biochemical systems is key to uncover the mechanisms that cells use to control their responses to environmental stimuli. In this work we use the time-integral of chemical species as a measure of a network's ability to cumulatively transmit signals encoded in spatiotemporal concentrations. We identify a class of nonlinear reaction-diffusion networks in which the time-integrals of some species can be computed analytically. The derived time-integrals do not require knowledge of the solution of the reaction-diffusion equation, and we provide a simple graphical test to check if a given network belongs to the proposed class. The formulae for the time-integrals reveal how the kinetic parameters shape signal transmission in a network under spatiotemporal stimuli. We use these to show that a canonical complex-formation mechanism behaves as a spatial low-pass filter, the bandwidth of which is inversely proportional to the diffusion length of the ligand.