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Say I have some polynomial $p(x)$ and want to express its $n$th integral, is there a closed form for this?

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    @Pedro You can answer your own question.2013-03-16

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Noticing that for $d\leqslant p$, we have that the $d$-th derivative of $x^p$ is $x^{p-d}\frac{p!}{(p-d)!},$ we get what Srivatsan wrote in his comment, that is, the $n$-th integral of $x^r$ is $\frac{r!}{(r+n)!}x^{r+n}$ (take $p-d=r$ and $d=n$) up to a polynomial of degree $\leqslant n-1$ (as at each integration, we need to integrate the constant of the previous one). Then the result follows by linearity.

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    Or just take $d$ to be negative. Remember that negative iteration of a function or operator is iteration of the inverse, so differentiating a negative number of times is equivalent to integrating a positive number of times.2014-09-14