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When parameters to a integral are integers, I often fall back to calculating the "general" result, make a table where the parameter is explicitly given integer values, and then feed to FindSequenceFunction. In this case: table = Table[h[x, a, e] - g[x, a, e], {a, 1, 11}]; limits = Limit[#, e -> 0] & /@ table; f = FindSequenceFunction[limits]; ...


The simplest method is to wrap xdummy in Quantity, e.g. result = Integrate[Quantity[3, "kN/m"] (1 - x/Quantity[10, "m"]), {x, Quantity[xdummy,"m"], Quantity[10, "m"]}] (* Quantity[(15000 - 3000 xdummy + 150 xdummy^2), ("Kilograms" "Meters")/("Seconds")^2] *) Since xdummy is now effectively unitless, you no longer need to include units in Plot, ...

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