Let $\alpha$ be multi-index. Consider the iterated integral $$\int_{a_n}^{b_n} \cdots \int_{a_1}^{b_1} \left(D^{\alpha}f(x)\right)g(x) \, \mathrm{d}x_1 \ldots \mathrm{d}x_n.$$ Does anyone know a slick formula for this, moving all derivatives from $f$ to $g$? Of course, this is just iterated integration by parts, but doing this by hand seems like a huge pain. A simple reference would be sufficient, thanks!


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