Ian, that's not a "first derivative index" in the second equation. The R' in R(l) is just there to distinguish it from the R in R(v) - i.e. to show that the function R'() takes a different format from R().

The reason for using integration (the "peculiar formula containing infinities" as you call it!) is, as Alberto says, because you want to make sure that the TOTAL energy is the same. ( R'(l) has a different SHAPE from R(v), but the total AREA under either curve must be the same. )

Just replacing v/c by l in the first equation does NOT give you "a function in l", i.e. you can replace v/c by l in the right hand side of the first equation, but you can't just change the left hand side from R(v) to R'(l). (I think that would work if "l was linear in v" - i.e. if l = v + constant, say - but not when there's a reciprocal relation between l and v. Which ties in with Alberto's comment about the derivative being the wrong sign, of course.)