# How to integrate x^x?

• Apr 25th 2009, 04:52 PM
chengbin
How to integrate x^x?
I'm bored today so I tried to integratel $x^x$, but I couldn't. How do you integrate that?
• Apr 25th 2009, 04:56 PM
Chop Suey
You can't express it in elementary functions.
• Apr 25th 2009, 04:59 PM
chengbin
Quote:

Originally Posted by Chop Suey
You can't express it in elementary functions.

What does that mean? What is elementary functions? So does that mean you can express it in another function?

Also, the derivative of $x^x$ is $x^x(\ln x+1)$

How would you integrate that if you don't know that formula?
• Apr 25th 2009, 05:05 PM
Krizalid
it means that you'll never find a certain function whose derivative yields the integrand.
• Apr 25th 2009, 06:24 PM
mr fantastic
Quote:

Originally Posted by Krizalid
it means that you'll never find a certain function whose derivative yields the integrand.

Actually it's not difficult. It's just that nobody bothered to think of defining a special function to account for its antiderivative.

There is a solution - the function F such that dF/dx is x^x. This function is also known as the Fantastic F-function.

It can't be written anymore nicely than that. There is nothing surprising about this. Almost no functions have integrals that can be written out nicely and explicitly in some closed form.
• Apr 25th 2009, 08:51 PM
curvature
Quote:

Originally Posted by chengbin
What is elementary functions?

See this for elementary functions:
Elementary function - Wikipedia, the free encyclopedia