Considering a method to find the anti-derivative of an (sufficiently smooth) real function by differentiating published some years ago (equation (48) in Kempf et al., New Dirac Delta function based methods with applications to perturbative expansions in quantum field theory): \begin{equation} \int^x f(x')\,dx' = \lim_{y \to 0} f\left(\frac{\partial}{\partial y}\right) \frac{\mathrm{e}^{xy}-1}{y} +C, \end{equation} I'm wondering whether Euler in his very imaginative calculations (to say the least) did use some techniques (in special cases) that amount to this formula.
Any hints are welcome.