Computing $\int \sqrt{x\sqrt[3]{x\sqrt[4]{x\sqrt[5]{x\cdots}}}} \,\mathrm{d}x$

sequences-and-series definite-integrals

$\sqrt{x\sqrt[3]{x\sqrt[4]{x{...\sqrt[n]{x}}}}} = \sqrt{x} \cdot \sqrt{\sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{\sqrt[4]{x}}} \cdot... = \displaystyle{x^{\sum_{k=2}^n\frac{1}{k!}}}$.

Let $r = \displaystyle{{\sum_{k=2}^n\frac{1}{k!}}}$. So $\displaystyle{\int{x^rdx} = \frac{x^{r+1}}{r+1} + \mathcal{C}}$.

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