# Thread: How to integrate x^x?

1. ## 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?

2. You can't express it in elementary functions.

3. 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?

4. it means that you'll never find a certain function whose derivative yields the integrand.

5. 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.

6. Originally Posted by chengbin
What is elementary functions?
See this for elementary functions:
Elementary function - Wikipedia, the free encyclopedia