I can't find this integral around here, does anybody suggest how to calculate this integral?
$$ I = \int x^x dx. $$
Thanks in advance
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2Probably no closed form – Sep 17 '14 at 11:08
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Why do you suppose that a closed formula exists? – Siminore Sep 17 '14 at 11:08
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use $y=x^x$ (there is no closed form) – ulead86 Sep 17 '14 at 11:09
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If there is a closed form, you'll probably get there by writing $$x^x = \exp(x \ln x)$$ (this is how you can find it's derivative fairly easily) – AlexR Sep 17 '14 at 11:20
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Concerning a closed form this thread may help. Approximations may be obtained using AlexR's comment as shown by Wolfram Alpha. – Raymond Manzoni Sep 17 '14 at 11:44
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This integral does not have a solution in terms of elementary functions. However a series approximation yields $$ \int x^x\ dx= \int e^{\ln x^x}\ dx = \int \sum_{k=1}^{\infty} \frac{x^k\ln^k x}{k!}\ dx $$
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