Why is it difficult to integrate x^x

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Discussion Overview

The discussion revolves around the challenges of integrating the function x^x, exploring why it cannot be expressed in terms of elementary functions. Participants also touch on related integration problems and the implications for calculus education.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • Some participants assert that x^x cannot be integrated in terms of elementary functions, emphasizing that most functions do not have simple antiderivatives.
  • Others propose that defining a special function for the antiderivative of x^x could be a way to study its properties.
  • A participant mentions that numerical techniques can approximate definite integrals of x^x to arbitrary accuracy.
  • Some express frustration with integration tools like Mathematica, which do not provide a solution for x^x in terms of complex functions.
  • There are discussions about the educational implications of presenting only "nice" integration problems in calculus courses, leading to misconceptions about the nature of functions and their integrals.
  • One participant suggests using Taylor expansion as a method to approach the integration of x^x.
  • Another participant mentions the possibility of defining a function F such that its derivative is x^x, but notes that it cannot be expressed in a closed form.
  • There are references to other integration problems, such as integrating x*Sec(x), with varying levels of success and understanding among participants.

Areas of Agreement / Disagreement

Participants generally agree that x^x cannot be integrated in terms of elementary functions, but there are multiple competing views on how to approach the problem and the implications for calculus education. The discussion remains unresolved regarding the existence of a defined function for the antiderivative of x^x.

Contextual Notes

Some participants mention the limitations of integration tools and the lack of defined special functions for certain integrals. There is also a recognition of the complexity involved in integrating functions that appear "nice."

  • #91
Дьявол said:
Thanks for the correction everybody.

Here is my new solution.

If
<br /> x^x = e^{x\ln x}<br />
then

As a side for some reason I was thinking about a good series representation.

If I expand ln(x) first then one gets

x^x=\Pi_{n=0}^{\infty}exp \left( \frac{(-x)^n}{(n+1)!} \right)

Consider what happens now if you plug in \frac{(-x)^n}{(n+1)!} for the Taylor series of an exponential. And then multiply each of these terms together. If you are only interested in the first N terms of the Taylor series, then you only need to consider the first N terms in the product.

Also looking at the series, it kind of looks like the logarithm of x^x is \frac{1}{x}exp(-x)

but I think I must have made a mistake.
 
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  • #92
Guys, this thread was from 2005. Is this normal?
 
  • #93
Pinu7 said:
Only possible if we can define and inverse function of x^x(which will only work for x>0).

And then, I doubt the integral COULD be solved analytically. However, maybe it would be simpler to approximate?
Hallo Pinu7
It seems we are answering a question outdated. Do you matter?, I don't. Mathematical problems survived wars, so why not years? Ok how to integrate x^x?
You say you can inverse the function for x>0. What will that be?
greetings Janm.
 
  • #94
Well the best I managed is an infinite series, which curiously when evaluated from 0 to 1 gives
[[math]]
1-\frac{1}{2^{2}}+\frac{1}{3^{3}}-\frac{1}{4^{4}}+...
[[math]]
I wrote x^x as e^xlnx and used the infinite series e^x. Integrating term by term and used l'Hospital's rule and recursive nature of the integrals to generate the sequence above. Checked using definite integral calculator
 
  • #96
  • #97
how integrate x^x^x ?my teacher ask me)what do you think about it?
 
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  • #98
Alejandroman8 said:
how integrate x^x^x ?my teacher sak me)what do you think about it?

I think you're in for a lot of swearing and bloody knuckles. Since x^x is positive for all x > 0 then x^{(x^x)} is continuous for x > 0 and therefore Riemann integrable. But trying to find any sort of friendly resolution to it is a fool's errand.

Numeric approximation is your best hope since I think even power series will be intractable.

--Elucidus
 
  • #99
Alejandroman8 said:
how integrate x^x^x ?my teacher ask me)what do you think about it?

Let's not think about how to integrate, just think about the easy case, differentiation, you still wouldn't get nice results at all.

http://the-genius-group-from-uc-berkeley.googlegroups.com/web/Tetration%20Differentiation.pdf?gsc=JivL3wsAAABzdSJOyPIJbvk4ERCPyg5o
 
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  • #100
Basically, I say (cause and effect stuff/ quark stuff / physically imposside to divide stuff / blah b

going along with everything in nature can be defined as and affacting the reality as (itself is) ,,, call it single definable existence or basically the law of of cause and affect exists...: blah blah blah, no fractions or partial exitstance. (avoid s domain)

(according to the equation derivative guy, (tesla I like)... nueton (sp)...) slope!

der x to x is
is
(xtox - ((xex) - (xe1)to(xe1)) / 1

rise over run in a a non fraxctional (cause and effect world) tadah...

Take a microprossessors class in Electrical Engineering (learn machines do math etc... O'Mally Universiy of Fl... nothing better, I've seen) ...followed by a deep physicall class in statistics (math side, if it exists or maybe not (math dept, might kick your but in statistics) do stilll calll it "counting statistics? get into those sum equations/lin equations and don't get they are the same,
take a math department lin equations classes (at the same time)

that semester will be fun...
 
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