Any formal methods to evaluate $\int x^x\mathrm dx$

Asked Dec 25 '22 at 09:06

Active Dec 25 '22 at 09:33

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Are there any formal methods to evaluate $I$ ?

$$I = \int x^x\mathrm dx$$

Online calculators don't give any output. So, are there any known methods to evaluate at least some definite integrals like $\int_1^2 x^x\mathrm dx$ , $\int_1^ex^x\mathrm dx$ or any ?

edited Dec 25 '22 at 09:12

Angelo

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asked Dec 25 '22 at 09:06

An_Elephant

2,704

2

See sophomore's dream from Wikipedia. – soupless Dec 25 '22 at 09:14
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You can approximately find them by using $x^x$'s Taylor expansion: https://math.stackexchange.com/questions/802256/the-nth-derivative-of-xx/4605115#4605115 – Bob Dobbs Dec 25 '22 at 09:39
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This has no closed form antiderivate, you have to compute a definite integral numercially. – Peter Dec 25 '22 at 09:48
@soupless I thought for a neat technique atleast for the definite integral though it isn't now. Thanks very much. – An_Elephant Dec 25 '22 at 11:03
Does this answer your question? Finding $\int x^xdx$ – Тyma Gaidash Dec 25 '22 at 13:10
@TymaGaidash ( I asked for any techniques for definite integral while the question linked is only of indefinite integral). Despite it's high votes, I don't know why it didn't came as similar question while I was posting and ensuring to avoid duplicacy. – An_Elephant Dec 25 '22 at 14:36

Any formal methods to evaluate $\int x^x\mathrm dx$

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