how Can I Integrate X^X?

In terms of elementary functions, you can't.

how Can I Integrate X^X?

\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 :p

What's That Answer Supposed To Be?

how Can I Integrate X^X?

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.

What's That Answer Supposed To Be?

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.

how Can I Integrate X^X?

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

IOW, make \xi(x) itself an elementary function.
But my interpretation may be wrong.
The term "elementary function" is well-defined. From http://en.wikipedia.org/wiki/Elementary_functions:

In mathematics, an elementary function is a function built from a finite number of exponentials, logarithms, constants, one variable, and roots of equations through composition and combinations using the four elementary operations (+ – × ÷). The trigonometric functions and their inverses are assumed to be included in the elementary functions by using complex variables and the relations between the trigonometric functions and the exponential and logarithm 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.

The term "elementary function" is well-defined. From http://en.wikipedia.org/wiki/Elementary_functions:

Thanks! I did not know that.

Or forgot -- it's been a long time since B.S.

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

If It Is Not Possibe, Then How Can I Find Out The Area Of That Curve?

Why Do You Type This Way?

If It Is Not Possibe, Then How Can I Find Out The Area Of That Curve?

Plot the function and integrate it numerically.

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.

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?

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

I don't see how x^x = x^3?

I'm not familiar with the notation . I read xsquared x
Oops