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Question Number 216486 by Tawa11 last updated on 08/Feb/25

$${\int }_0^1\mathrm{x}\sqrt{\mathrm{x}\sqrt[3]{\mathrm{x}\sqrt[4]{\mathrm{x}\sqrt[5]{\mathrm{x}...}}}}\mathrm{dx}$$

Answered by mehdee7396 last updated on 09/Feb/25

$$x\times x^{\frac{1}{2}}\times x^{\frac{1}{6}}\times x^{\frac{1}{24}}\times x^{\frac{1}{120}}\times ...$$$$=x^{\underset{1}{\sum ^{\mathrm{\infty }}}\frac{1}{n!}.}=x^{e-1}$$$${\Rightarrow I={\int }_0^1{x}^{e-1}dx=\frac{1}{e-1}\left(x^{e-1}\right)]}_0^1$$$$=\frac{1}{e-1}$$$$$$

Commented by Tawa11 last updated on 09/Feb/25

$$\mathrm{Thanks}\mathrm{sir}.$$

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