Integration by parts. Push It to the Limit

Question

Given a function $f$, a primitive $F$ is given by $$ F(x)=\int\,f(x)\,\mathrm{d}x. $$ Using repeated integration by parts it seems that one can rewrite $F$ as $$ F(x)=xf(x)-\frac{1}{2!}x^2f'(x)+\frac{1}{3!}x^3f''(x)-\frac{1}{4!}x^4f'''(x)+\cdots+(-1)^{k+1}\frac{x^{k+1}}{(k+1)!}f^{(k)}(x)+\cdots $$ or as \begin{align*} F(x)&=\Bigg[\sum_{k\geq0}(-1)^{k+1}\frac{x^{k+1}}{(k+1)!}\partial_x^k\Bigg]f(x)\\ &=\frac{e^{-x\partial_x}-1}{\partial_x}f(x). \end{align*} Does this have some utility, application or use somewhere?