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I have searched on various websites that it does not seem possible to integrate x^x? But my friend Wesley said that x^x could be integrated by using puiseux series. I have looked up puisex series on Wikipedia, but it only says it is a generalization of the power series, but does not mention how it is related to integrating x^x? Could anyone help me clarify my doubts on how puiseux series is related to integration of x^x?

Wikipedia Link:https://en.wikipedia.org/wiki/Puiseux_series

Jackson Ma
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1 Answers1

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To answer the question, this function can not be integrated in terms of elementary functions. But:

$$\int x^x\space\text{d}x= \int\exp\left(\ln\left(x^x\right)\right)\space\text{d}x= \int\sum_{\text{k}=1}^{\infty}\frac{x^\text{k}\ln^\text{k}\left(x\right)}{\text{k}!}\space\text{d}x=\sum_{\text{k}=1}^{\infty}\frac{1}{\text{k}!}\int x^\text{k}\ln^\text{k}\left(x\right)\space\text{d}x$$

Jan Eerland
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