The Integral of (ln(x))^x dx seems easy enough but to express it indefinitely?

Question

pawanyadav · Answer

Isn't it ln(x^x)dx ...

ganeshie8 · Answer

could you provide some context of this problem ?

kainui · Answer

Might just be a coincidence, but I noticed these graphs seem to be approximately the same: $$(\ln x)^x \approx e^{x-e}$$ https://www.desmos.com/calculator/dsxiahpiw6 Of course one of these is much easier to integrate than the other, so it would be nice if there was some definite relationship between the two.

miracrown · Answer

Shouldn't it be in(x^x)dx @daniel.ohearn1

miracrown · Answer

We can use a property of logarithms
xln(x)dx
we pull down the exponent

miracrown · Answer

any ideas about how to integrate xln(x) ?

miracrown · Answer

so we need to assign u and dv... what could we let u be ?

bobo-i-bo · Answer

If it is (lnx)^x, then note that (lnx)^x=e^(x*ln(lnx))

miracrown · Answer

Yes, I agree with that.

agent0smith · Answer

Something tells me the integral of (ln x)^x is not "easy enough": http://m.wolframalpha.com/input/?i=integral+of+%28ln+%28x%29%29%5Ex&x=0&y=0

daniel.ohearn1 · Answer

No it isn't ln(x^x) nor is it "impossible".   
But it could be unique, and there could be a umbrella function for it..

bobo-i-bo · Answer

It's convincing enough for me :P