# Integrate x^x

1. Sep 30, 2005

### Castilla

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

2. Sep 30, 2005

### amcavoy

Not that my TI-89 knows of.

3. Sep 30, 2005

### 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

Last edited: Sep 30, 2005
4. Oct 2, 2005

### Castilla

Thanks to both of you.

Castilla.

5. Oct 4, 2005

### saltydog

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.

Last edited: Oct 4, 2005
6. Oct 20, 2005

### benorin

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.

7. Oct 20, 2005

### 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

8. Oct 20, 2005

### 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"

9. Oct 30, 2005

### Johnny Numbers

Something tells me that might make it worse.