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.
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