How to integrate the nth root of n

Question

I am working on a problem in which there is an integral containing $x^\frac1x$. I have looked at this question for some help: Behavior of the Nth root of N?

I have changed $x^\frac1x$ to $e^{ln(x)/x}$ but I still have not found a way to integrate it. How can I integrate either equation? Thanks. If you vote down, please tell me why so I can improve this question.

Wolfram alpha say "no result found in terms of standard mathematical functions", so there is probably no easy way to do this. What's your problem exactly, is your integral definite or indefinite? — Tryss, Feb 24 '15 at 14:26
@Tryss The integral is definite(but inside a limit). I have already tried many of the online function integrators out there and none have given me an answer. — Progo, Feb 24 '15 at 14:29
You may have more luck in posting the whole expression to calculate. Maybe only an asymptotic growth or an approximation is necessary for your problem — Tryss, Feb 24 '15 at 14:31
As said, it would be better to know what is the integral. Please, post it. — Claude Leibovici, Feb 24 '15 at 14:37
$\int x^{1/x},dx$ is not an elementary function. Some (indeed most) integrals are like that. — GEdgar, Feb 24 '15 at 14:39
@Progo: Only if the bottom $x$ is the variable, and the top $x$ is constant. — GEdgar, Mar 07 '15 at 14:27
@GEdgar Then is can this function not be integrated? Doesn't every function have an anti-derivitive? — Progo, Mar 07 '15 at 14:30
@GEdgar I basing that guess off of this: https://answers.yahoo.com/question/index?qid=20100201165158AAlAqJy — Progo, Mar 07 '15 at 14:32
It has an antiderivative, as does every continuous function. But (as I said) this antiderivatvie is not an elementary function. — GEdgar, Mar 07 '15 at 14:32
In that yahoo answer, the $n$ in the exponent is constant. So then we can easily do the integral! — GEdgar, Mar 07 '15 at 14:33
@GEdgar Do you have any suggestions for where I might start looking for the anti-derivitive of this function? — Progo, Mar 07 '15 at 14:51

score 2 · Accepted Answer · answered Mar 07 '15 at 14:56

2

Define $$ F(x) = \int_{1}^{x} s^{1/s}\;ds,\qquad x>0 $$ Then $F'(x) = x^{1/x}$ so $F$ is an antiderivative. There is no simpler way to write this.

answered Mar 07 '15 at 14:56

GEdgar

111,679

So, x^(1/x) is the anti-derivitive of x^(1/x)? – Progo Mar 07 '15 at 15:03
No that function $F$ is the antiderivative of $x^{1/x}$. – GEdgar Mar 07 '15 at 15:05
Ok, thanks! This is very helpful! – Progo Mar 07 '15 at 15:06

How to integrate the nth root of n

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