Integrate X^X.

Discussion in 'Physics & Math' started by manoharprabhu, Jan 22, 2008.

1. manoharprabhuRegistered Member

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how Can I Integrate X^X?

3. D HSome other guyValued Senior Member

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In terms of elementary functions, you can't.

5. AbsaneRocket SurgeonValued Senior Member

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$\int_1^n x^{x}dx = \xi(n)$

Of course, this is MY definition. If you are looking for a different solution, I suggest you stop looking for one

7. manoharprabhuRegistered Member

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What's That Answer Supposed To Be?

8. LetticiaRegistered Senior Member

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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.

9. LetticiaRegistered Senior Member

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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.

10. Crunchy CatF-in' *meow* baby!!!Valued Senior Member

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

11. D HSome other guyValued Senior Member

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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.

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13. manoharprabhuRegistered Member

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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

14. manoharprabhuRegistered Member

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

15. D HSome other guyValued Senior Member

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Why Do You Type This Way?

16. paulfrRegistered Senior Member

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Plot the function and integrate it numerically.

17. manoharprabhuRegistered Member

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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?

18. NugletsRegistered Member

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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.

19. MylesRegistered Senior Member

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

20. NugletsRegistered Member

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

21. MylesRegistered Senior Member

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