\(\int_1^n x^{x}dx = \xi(n)\) Of course, this is MY definition. If you are looking for a different solution, I suggest you stop looking for one Please Register or Log in to view the hidden image!

You can't. Antiderivative of x^x is not reducible to any elementary functions -- which is not at all unusual. This is true of MOST functions. Calculus books usually list antiderivatives of a hundred or so rather simple functions, and toward the end of the list the antiderivatives get really bizarre. It does not take much effort to come up with a function which can not be precisely integrated. x^x is one.

My understanding is that nothing prevents you from giving a name, such as \(\xi(x)\), to the function defined as antiderivative of \(x^{x}\), and then studying its properties. IOW, make \(\xi(x)\) itself an elementary function. But my interpretation may be wrong.

I think Letticia might be right; however, just in case you might want to experiment with integration by parts and the exponent variable.

The term "elementary function" is well-defined. From http://en.wikipedia.org/wiki/Elementary_functions: That does not stop you from defining the function \(\xi(x)\equiv\int_0^xt^tdt\). Mathematicians do this all the time. Those functions that have a widely agreed-upon name and definition are the "special functions". Many of the elementary functions are "special", but only a handful of the special functions are elementary. The integral of \(x^x\) is not elementary and it is not of much use practical or impractical use (yet!), so it is not particularly "special", either.

Can I Integrate It By Parts By Taking X^X As First Function, And 1 As Second Function? i.e----- Integration Of (X^X)*1

Wait A Minute.... Differentiation Of X^X is X^X(1+log(x)). That Should Mean That Integration Of X^X(1+log(x)) Is Equal To X^X. Am I Correct?

That is not the same as integrating x^x though. Many times the functions that appear to be more difficult are in fact easier/possible to integrate.

50 years ago I would have said that x^x = x cubed which would differentiate as 3 x^. Where an I going wrong ?