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](https://arxiv.org/pdf/1404.0747.pdf)):
\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.



Did Euler know (unconsciously) to integrate by differentiating?
I don't know about that particular integral, but Euler certainly knew about integrating by differentiating. He wrote about it in his *Exposition de quelques paradoxes dans le calcul integral* (1758). A recent summary of that work can be found in


A. Fabian and H.D. Nguyen,
Paradoxical Euler: integrating by differentiating,
*The Mathematical Gazette* **97** (2013), no. 538, 61-74.


