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I think it doesn't matter if the reaction is exothermic or endothermic because those are the final states while Kinetics focuses on the transition between reactants to products. Increasing the temperature will then increase the rate constant!
Raising the temperature for endothermic reactions will make the production of products more favorable (forward reaction). Conversely, raising the temperature for exothermic reactions makes the production of products less favorable (forward reaction). As for temperature's effect on rate constants, you can generalize trends based on the Arrhenius equation; higher temperature results in higher values for rate constants.
Since the equilibrium constant for a reaction is equal to the ratio of the rate constants for the forward and reverse reactions, and since the equilibrium constant is increasing for the forward reaction due to it being endothermic, that means the rate constant for the forward reaction is increasing as well.
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