Can the Lambert-W Function Solve the Integral of x^x?

[Castilla]

Hello, a question: is there a reasonable way to obtain $$\int x^xdx$$ ??

 

[amcavoy]

Not that my TI-89 knows of.

 

[lurflurf]

The integral cannot be expressed in elementary functions.
If an infinite series will do
$$\int \ x^xdx=\int \ e^{x\log(x)}dx=C+\sum_{k=0}^\infty \ \int_0^x \ \frac{\log^k(t)}{k!}t^kdt$$
so if an infinite sum will do find an easy integral gets you a nice one.
$$\int \ x^{-x}dx$$
is similar
$$\int_0^1 x^xdx$$
and
$$\int_0^1 x^{-x}dx$$
are extra nice

 

[Castilla]

Thanks to both of you.

Castilla.

 

[saltydog]

Thanks to both of you.

Castilla.

Yea thanks.

Is there a way to remove the integral sign? Looks like they can be analytically determined. for example:

$$\int x^2ln^2(x)dx=2/3 x^3-2/9 x^3ln(x)+1/3 x^3ln^2(x)$$

and higher powers involve corresponding higher powers of x and ln(x) in the antiderivative.

 

[benorin]

Hello, a question: is there a reasonable way to obtain $$\int x^xdx$$ ??

It is provably impossible to represent that antiderivative as a finite combination of elementary functions. See the bottom of http://mathworld.wolfram.com/Integral.html for a confrimation of this fact.

This is not to say that it does not have a solution, it just not pretty.

 

[dephys]

The Problem could be solved by using the
integral of the infinite series
which could be calculated by the numerical methods
if u are interested in the solution i may work out the
algorithm or program for You

 

[SebastianG]

Idea

My intuition tells me you can use the Lambert-W function on this one. Just as Eisenstein made it work for "power tower" functions (N^N^N^N^N^N^N...). It might work.

If you want to know about that function, check the link on the post "A very interesting question about Complex Variable"

 

[Johnny Numbers]

My intuition tells me you can use the Lambert-W function on this one. Just as Eisenstein made it work for "power tower" functions (N^N^N^N^N^N^N...). It might work.
If you want to know about that function, check the link on the post "A very interesting question about Complex Variable"

Something tells me that might make it worse.