We derive the spatially resolved star formation history (SFH) for a 96deg^2^ area across the main body of the Large Magellanic Cloud (LMC), using the near- infrared photometry from the VISTA survey of the Magellanic Clouds (VMC). The data and analyses are characterized by a great degree of homogeneity and a low sensitivity to the interstellar extinction. 756 subregions of size 0.125deg^2^ - corresponding to projected sizes of about 296x322pc^2^ in the LMC - are analysed. The resulting SFH maps, with typical resolution of 0.2-0.3dex in logarithm of age, reveal main features in the LMC disc at different ages: the patchy star formation at recent ages, the concentration of star formation on three spiral arms and on the Bar up to ages of ~1.6Gyr, and the wider and smoother distribution of older populations. The period of most intense star formation occurred roughly between 4 and 0.5Gyr ago, at rates of ~0.3M_{sun}_/yr. We compare young and old star formation rates with the observed numbers of RR Lyrae and Cepheids. We also derive a mean extinction and mean distance for every subregion, and the plane that best describes the spatial distribution of the mean distances. Our results cover an area about 50 per cent larger than the classical SFH maps derived from optical data. Main differences with respect to those maps are lower star formation rates at young ages, and a main peak of star formation being identified at ages slightly younger than 1Gyr.