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We use cosmological hydrodynamical simulations to assess the feasibility of constraining the thermal history of the intergalactic medium during reionisation with the Lyman-alpha (Lya) forest at z=5. Integrated heating impacts on the transmitted flux power spectrum, which results in measurable pressure smoothing that can be isolated from Doppler broadening at this redshift. We parameterise integrated heating using the cumulative energy per proton, u_0, deposited into a gas parcel at the mean background density, a quantity that is linked pressure smoothing and the gas density power spectrum in the simulations. We construct mock observations of the line of sight Lya forest power spectrum and use a Markov Chain Monte Carlo approach to recover u_0 at redshifts 5