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Question Number 160204 by amin96 last updated on 25/Nov/21

$${\int }_0^1\frac{\mathbf{x}}{1-{\mathbf{x}}^3}=?$$

Commented by mr W last updated on 25/Nov/21

$$ithink{\int }_0^1\frac{x}{1-x^k}dxdiverges.(k⩾1)$$

Answered by Ar Brandon last updated on 25/Nov/21

$$I={\int }_0^1\frac{x}{1-x^3}dx=\frac{1}{3}{\int }_0^1{x}^{-\frac{1}{3}}{(1-x)}^{-1}dx$$$$=\frac{1}{3}\beta (\frac{2}{3},0)=\frac{1}{3}\cdot \mathrm{\Gamma }\left(0\right)\mathrm{divergent}\mathrm{since}$$$$\mathrm{\Gamma }\left(x\right)\mathrm{diverges}\mathrm{for}x\in {\mathbb{Z}}^{-}$$

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