definite and indefinite sums and integrals

It just occurred to me that I tend to think of integrals primarily as indefinite integrals and sums primarily as definite sums. That is, when I see a definite integral, my first approach at solving it is to find an antiderivative, and only if that doesn't seem promising I'll consider whether there might be a special way to solve it for these special limits; whereas when I see a sum it's usually not the first thing that occurs to me that the terms might be differences of some function. In other words, telescoping a sum seems to be just one particular way among many of evaluating it, whereas finding antiderivatives is the primary way of evaluating integrals. In fact I learned about telescoping sums much later than about antiderivatives, and I've only relatively recently learned to see these two phenomena as different versions of the same thing. Also it seems to me that empirically the fraction of cases in which this approach is useful is much higher for integrals than for sums.

So I'm wondering why that is. Do you see a systematic reason why this method is more productive for integrals? Or is it perhaps just a matter of education and an "objective" view wouldn't make a distinction between sums and integrals in this regard?

I'm aware that this is rather a soft question, but I'm hoping it might generate some insight without leading to open-ended discussions.

Your answer is probably buried within this statement: "I learned about telescoping sums much later than about antiderivatives."

All mathematicians, and a substantial fraction of college graduates, study integration for at least a year, frequently many more. Rather fewer study discrete techniques, and only specialists study difference calculus for a year or more. The analog to finding an antiderivative is finding a WZ pair, which is viewed as an "advanced technique".

Perhaps if we were to reverse the curriculum, teaching everybody differences and reserving differential calculus for specialists, your question might have been reversed. Incidentally, I think this might not be such a bad idea; continuous methods, which used to rule the world of science and engineering, are rapidly being replaced by discrete approximation and simulation.

Also, read about how Feynman learned some non-standard methods of indefinite integration (such as differentiating under the integral sign) and used these to get various integrals that usually needed complex integration.