How Can We Integrate x^x Effectively?

[thebetapirate]

$$\int x^{x}{d}x\x$$

What is it?

I have tried integration by parts and substitutions of various kinds and have arrived at certain solutions but none that look pretty.

 

[HallsofIvy]

My standard question for posts like this: Do you have any reason to think it has an elementary anti-derivative? ("Pretty" or not!)

 

[thebetapirate]

Then my follow-up question is this: if no anti-derivative exists, how do you prove that?

I've been working with $$\int e^{u}e^{ue^{u}}du$$ the derivation of which becomes apparent after the substitution of $$\ x=e^{u}$$.

 

[sutupidmath]

Then my follow-up question is this: if no anti-derivative exists, how do you prove that?

I've been working with $$\int e^{u}e^{ue^{u}}du$$ the derivation of which becomes apparent after the substitution of $$\ x=e^{u}$$.

Well, Halls is not saying that there is no antiderivative at all, it is just that the antiderivative will not be in terms of elementary functions. In other words, the antiderivative will probbably include a gamma, zeta, gauss etc function in it!

 

[thebetapirate]

Argh, so when I wrote anti-derivative in my response post I actually meant anti-derivative in terms of elementary functions. Again, how would that be proven?

 

[John Creighto]

I'm tempted to try applying the limit deffinition of the integral.

 

[John Creighto]

Okay maybe it's futile but I'll start it anyway.

$$\int_{0}^{x}x^xdx=\mathop {\lim }\limits_{N \to \infty } \sum_{n=1}^{N \ x}(n/N)^{n/N}$$

Okay, wikipedia is going slow so I'll see if I can get further tomorrow.

 

[Gib Z]

Determining weather or not an integral is expressible in terms of elementary functions without actually calculating the integral is somewhat complex. The Risch algorithm is sometimes used, but I don't know too much about it.

 

[Big-T]

http://www.sosmath.com/calculus/integration/fant/fant.html

 

[John Creighto]

http://www.sosmath.com/calculus/integration/fant/fant.html

Those names theorems seem pretty useful.

 

[johndoe]

Calling an integral elementary means it can be integrated by simple basic methods? like int e^x=e^x

Sorry not a native speaker.

 

[Gib Z]

Well, not exactly. Calling an integral "elementary" is a matter of opinion, but for an integral to be expressible in terms of elementary functions means that the anti-derivative is composed of a finite sum/product of elementary (basically the simple functions we learn about in high school and undergrad courses) functions. A proper list can be found on Wikipedia. There are many integrals that can be solved in terms of elementary functions that are still quite hard to do lol.