• #### what is integral of x^x = ?

From drhuang57@gmail.com@21:1/5 to All on Sat Feb 20 16:51:50 2021
what is integral of x^x = ?

mathHand.com

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• From Nasser M. Abbasi@21:1/5 to drhu...@gmail.com on Sat Feb 20 19:41:21 2021
On 2/20/2021 6:51 PM, drhu...@gmail.com wrote:
what is integral of x^x = ?

mathHand.com

A known trick is to write x^x as exp(x*ln x), then
use exp(u) = sum u^n/n!, n=0..infinity, and integrate
term by term, and sum. So you get

sum( integrate( (x*ln x)^n/n! ,x) , n=0..infinity)

For one term only, Rubi gives

oneterm = Int[ x^n*Log[x]^n/n!, x];

(Gamma[1 + n, -((1 + n) Log[x])] Log[x]^n (-((1 + n) Log[x]))^-n)/((1 + n) n!)

But this has no closed form sum

Sum[oneterm, {n, 0, Infinity}];

--Nasser

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• From Richard Fateman@21:1/5 to Nasser M. Abbasi on Sun Feb 21 10:54:50 2021
On Saturday, February 20, 2021 at 5:41:25 PM UTC-8, Nasser M. Abbasi wrote:
On 2/20/2021 6:51 PM, drhu...@gmail.com wrote:
what is integral of x^x = ?
........

But this has no closed form sum

Sum[oneterm, {n, 0, Infinity}];

--Nasser
on the other hand, for any particular integer n, you can integrate x^n*log(x)^n. (at least Maxima has no problem)
so you can generate as many terms as you wish, in elementary form, rational in x and log(x).
RJF

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