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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