This is a layman's answer. I'm by no means near an expert on integrals.
My first reaction to this question was that I am not sure that I agree and that since sums are integrals, so this does not even make sense, but then . Then I realized that the statement can be rephrased as
Continuous functions are easier to integrate than discontinuous ones
If we put it that way then perhaps the answer is
a) Almost anything is a lot of things are easier to do with a continuous function, or
b) in particular it is easier to find the indefinite integral of a continuous function than of a discontinuous one. Especially since the second one cannot have a nice anti-derivative (For instance it cannot be continuously differentiable for one).
Disclaimer This is a soft-answer and is not intended to contain mathematically rigorous statements or self-evident truths.