Consider a function f continuous on a compact interval.

Approximate it by a sequence of polygonal functions (you can).

Then consider a sequence of primitives of the polygonal functions (you can).

At last consider the limit of the latter sequence (you can).

Now you have found a primitive of f (you know) *without integration*.

This is the content of the first part of a not very known note by Lebesgue *Remarques sur la définition de l'intégrale*, Bull.Sci.Math. 29 (1905) 272-275 (see pdf for an exposition in English).

I doubt that such a thing was shown for the first time in 1905.

Lebesgue's good faith is beyond discussion of course.

Do you know something about ?

Newton integration. This allowed me the pun of the title. The second integration means Cauchy-Riemann integration of course. $\endgroup$ – Antonio Piciulin Apr 17 '13 at 7:45directlyavoiding the typical tour (i) develop the Riemann integral (ii) prove that continuous functions are Riemann integrable (iii) prove the fundamental theorem of the calculus for the Riemann integral. Obviously one can use step functions to approximate f. $\endgroup$ – Antonio Piciulin Apr 19 '13 at 7:28