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For an irreversible path, the integral of dQ/T is not equal to $\Delta S$. To get the change in entropy between the two end points of an irreversible path, you need to devise of a reversible path between the same two end points, and calculate the integral for that path. I doesn't matter what reversible path you choose; they will all give the same value for $\Delta S$. For more details on this, see the following link:
https://www.physicsforums.com/insights/grandpa-chets-entropy-recipe/ This link contains contains a simple recipe for determining the entropy change for an arbitrary irreversible processes, and presents several typical examples of irreversible processes encountered in thermodynamics courses, and how to apply the recipe to determine the change in entropy for each.