Exploring the Integration of x^x - A Puzzling Problem

• thebetapirate
In summary, the conversation discusses the concept of elementary anti-derivatives and how to determine if an integral can be expressed in terms of elementary functions. It also mentions the Risch algorithm and provides links to additional resources on the topic.
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.

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

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 $$\ x=e^{u}$$.

Last edited:
thebetapirate said:
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!

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?

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

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.

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 said:

Those names theorems seem pretty useful.

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

Sorry not a native speaker.

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.

1. What is the concept of x^x - A Puzzling Problem?

x^x - A Puzzling Problem refers to the mathematical expression x raised to the power of x, which is a common topic in calculus and algebra. It is often considered a puzzling problem because the result of this expression is not always easily determined and can lead to complex solutions.

2. What is the importance of exploring the integration of x^x - A Puzzling Problem?

Exploring the integration of x^x - A Puzzling Problem allows for a deeper understanding of mathematical concepts, such as limits, derivatives, and integrals. It also helps in developing problem-solving skills and critical thinking abilities. Additionally, this exploration can lead to the discovery of new mathematical techniques and applications.

3. What are the common strategies for solving x^x - A Puzzling Problem?

Some common strategies for solving x^x - A Puzzling Problem include using logarithms, substitution, and the binomial theorem. Other approaches may involve using calculus techniques such as L'Hopital's Rule and Taylor series. It is important to approach each problem with an open mind and try different strategies to find the most efficient solution.

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x^x - A Puzzling Problem has many real-life applications in various fields such as physics, biology, and economics. For example, it can be used to model population growth, compound interest, and the spread of diseases. It also plays a crucial role in understanding complex systems and phenomena in nature.

5. How can one improve their understanding of x^x - A Puzzling Problem?

To improve understanding of x^x - A Puzzling Problem, one can practice solving different types of problems and experiment with different techniques. Additionally, seeking help from a teacher or tutor, joining study groups, and exploring online resources can also aid in improving understanding. It is important to approach each problem with patience and persistence, as well as to continuously challenge oneself to think critically and creatively.

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