Is the ordinary integral a special case of the line integral?
If the field "defined" on the line actually satisfies the requirements to be a field, yes. Say, for example, if you're in C and choose your line to be R, then you're good. But if you choose the imaginary axis to be your line and do not change the definition of multiplication then you do not have a field.
But as for your question, I'm not sure. I think the two may be inherently different. If we're in R^2 the regular integral of any function along a line will be 0 because lines have measure 0, whereas it would be silly to have line integrals be 0 on all lines.
