Why is it difficult to integrate x^x

[latyph]

I tried doing this but could not,why is it so?

 

[dextercioby]

It's not difficult.Nobody thought of defining a special function to account for its antiderivative.

Daniel.

 

[Zurtex]

Because it can't be integrated in terms of elementary functions. Most functions are 'not easy' to integrate in this way.

 

[HallsofIvy]

Or, to say the same thing that Dextercioby and Zurtex said, in different words, because there is no elementary function whose derivative is x^x!

 

[arildno]

A better question would be:
Why is it simple to integrate f(x)=1?

 

[abia ubong]

i tried doing it ,but always get to a place i can't continue.who can help integrate [(x^x)(1+LOG[X])]^2.All help will be appreciated

 

[arildno]

i tried doing it ,but always get to a place i can't continue.who can help integrate [(x^x)(1+LOG[X])]^2.All help will be appreciated

You won't get any, since an anti-derivative of x^x is inexpressible in terms of elementary functions.

 

[goldi]

when i plugged it into the integrator of mathematica it gave it back as same...i don't know why it did not do computation.

 

[Zurtex]

when i plugged it into the integrator of mathematica it gave it back as same...i don't know why it did not do computation.

Because as said many times previously in this thread, it can not be integrated in terms of elementary functions.

 

[goldi]

i understand that yeah...but even Mathematica couldn't post the solution in terms of complex functions or whatever high level function it ocntains...
there must be a solution to it...What it is?

 

[Zurtex]

i understand that yeah...but even Mathematica couldn't post the solution in terms of complex functions or whatever high level function it ocntains...
there must be a solution to it...What it is?

There are no special functions defined in general mathematics for the integral.

If you want a function that is the anti-derivative of x^x then just define one and then you can study its properties.

 

[matt grime]

There is a solution it is the function F such that dF/dx is x^x. But we can't write it anymore nicely than that, and there is nothing surprising about it. Almost no functions have integrals that we can write out nicely and explicitly in some closed form. How many times must that be said in this thread? Shall we lock it now to stop yet another person having to say it?

 

[goldi]

Sorry but i am such a big fool that yours terminology is not clear to me...Last time-Has its integral ever calculated ...or as ppl are saying that it has such a function as integral that has never been defined./so is research going on over this

i have a question...how can we integrate x*Sec(x)
i have tried this question than any other question ever...
the point is that it was asked in my 12th class and when i plug it into Integrator i could not even understand the solution...

 

[arildno]

Any particular definite integral of x^x can, of course be calculated to an arbitrary degree of accuracy by numerical techniques.

 

[Jameson]

To integrate x*Sec(x) I would use integration by parts, but in this case the tabular method will work nicely.

$$\int x\sec{x}dx$$

$$\int udv = uv - \int vdu$$

$$u = x$$
$$dv = \sec{x}dx$$

That should get you started.

 

[dextercioby]

I think you meant

$$dv=\sec x \ dx$$

I'm not sure though...

Daniel.

 

[Icebreaker]

If you want a function that is the anti-derivative of x^x then just define one and then you can study its properties.

Has it been done before? Any interesting properties?

 

[Pyrrhus]

Look for Liouville's Principle about integration in finite terms.

 

[goldi]

To integrate x*Sec(x) I would use integration by parts, but in this case the tabular method will work nicely.

$$\int x\sec{x}dx$$

$$\int udv = uv - \int vdu$$

$$u = x$$
$$dv = \sec{x}dx$$

That should get you started.

that i would have had tried 100 times...after 1 step i am stuck and there is no way out...

 

[calculus1967]

i understand that yeah...but even Mathematica couldn't post the solution in terms of complex functions or whatever high level function it ocntains...
there must be a solution to it...What it is?

Well, there's F where
$$F(x) = \int_a^x t^t dt$$
and a can be any number greater than or equal to 0. Mathematica isn't able to find that.

 

[matt grime]

For those who think functions are *nice* (goldi etc) and aren't happy with our answers, can I ask what sin(x) is? I mean given x=2 radians say what is sin(x)? How do you define it? How do you find it? To me the answer is sin(2). There is nothing wrong with calculus's answer in the last post using the fundamental theory of calc. It is a very good function.

 

[Zurtex]

Has it been done before? Any interesting properties?

Attached below is a picture between x = 0 and x = 3 of the function:

$$f(x) = \lim_{a \rightarrow 0} \int_a^x t^t dt$$

 

[heman]

that i would have had tried 100 times...after 1 step i am stuck and there is no way out...

i checked into mathematica,,,it contains some functions of the form polylog but i think this is calculated with the knowledge of Complex Analysis.

 

[krab]

I tried doing this but could not,why is it so?

You could do it by Taylor expansion:

$$\int x^x dx=x + \frac{\left( -1 + 2\,\log (x) \right) \, x^2}{4} + \frac{\left( 2 - 6\,\log (x) + 9\,{\log (x)}^2 \right) \,x^3}{54} + \frac{\left( -3 + 12\,\log (x) - 24\,{\log (x)}^2 + 32\,{\log (x)}^3 \right) \,x^4}{768} + \frac{\left( 24 - 120\,\log (x) + 300\,{\log (x)}^2 - 500\,{\log (x)}^3 + 625\,{\log (x)}^4 \right) \,x^5}{75000} + \frac{\left( -5 + 30\,\log (x) - 90\,{\log (x)}^2 + 180\,{\log (x)}^3 - 270\,{\log (x)}^4 + 324\,{\log (x)}^5 \right) \,x^6}{233280} + {O(x^7)$$

 

[shmoe]

I wonder if students aren't done a disservice in first year calculus classes with their sterilized examples and problems. They'll be asked to do hundreds of integration problems, all rigged to work out nicely with the techniques they've just learned. Perhaps it will be mentioned that there are functions whose antiderivatives cannot be written in a "nice" form, but examples will be scarce- $$e^{-x^2}$$ being the stock one. After seeing such an unnatural ratio of nice examples to possibly one or two 'not-nice' ones, it's no surprise that many walk away believing themselves invincible and any function that itself looks 'nice' will have a 'nice' antiderivative waiting around the corner so they flap their arms around and bash their heads in frustration trying to find it. Makes me wonder if they bother to even consider why numerical techniques are taught at all?

 

[Zurtex]

I wonder if students aren't done a disservice in first year calculus classes with their sterilized examples and problems. They'll be asked to do hundreds of integration problems, all rigged to work out nicely with the techniques they've just learned. Perhaps it will be mentioned that there are functions whose antiderivatives cannot be written in a "nice" form, but examples will be scarce- $$e^{-x^2}$$ being the stock one. After seeing such an unnatural ratio of nice examples to possibly one or two 'not-nice' ones, it's no surprise that many walk away believing themselves invincible and any function that itself looks 'nice' will have a 'nice' antiderivative waiting around the corner so they flap their arms around and bash their heads in frustration trying to find it. Makes me wonder if they bother to even consider why numerical techniques are taught at all?

You are very much right, most people don't understand why we did a course in numerical analysis on my degree program. I think a lot will still have some very naive views on mathematics.

 

[heman]

$$e^{-x^2}$$
This is a nice question ,Our Professor explained it yesterday only and very much amused to find the way it was done.

 

[uart]

$$e^{-x^2}$$
This is a nice question ,Our Professor explained it yesterday only and very much amused to find the way it was done.

What do you mean "the way it was done", it has no antiderivative other than as definied by a special function. Or are you referring to the integration from -infinity to infinity, that's a different thing than the general antiderivative and can be done in closed form.

I wonder if students aren't done a disservice in first year calculus classes with their sterilized examples and problems. They'll be asked to do hundreds of integration problems, all rigged to work out nicely with the techniques they've just learned. Perhaps it will be mentioned that there are functions whose antiderivatives cannot be written in a "nice" form, but examples will be scarce- LaTeX graphic is being generated. Reload this page in a moment. being the stock one.

Yep I really know exactly what you mean. I can remember when I was first learning this stuff and when exp(-x^2) was introduced it was almost like it was this bizarre pathological function just because it didn't have a nice antiderivative.

 

[saltydog]

I tried doing this but could not,why is it so?

Can I introduce Philosophy here or will you guys beat-up on me? And I don't wish to make light of all the nice responses above also. Here goes:

Why does there exist some functions not integrable in closed form? It reduces to I think, to the question of why are some problems harder than others to solve? I mean, can there be a Universe with just easy nice problems: A Universe with the absence of any functions which can't be integrated in closed form?

Perhaps, but I don't think such a world would give rise to us. So Latyph, my efforts to answer your question is this: It's difficulty to integrate it is a reflection of the type of Universe that gives rise to an intellect that can ponder the question.

Edit: I should say "some functions whose antiderivative cannot be expressed in terms of simple functions or operations" .

 

[arildno]

I don't agree, saltydog:

Integration is "basically" a task of summing up an infinite amount of individual contributions; i.e, at the very outset a Herculean task no one in their right minds would assume could actually ever be completed in an exact manner.
Thus, the basic question is rather:
Why are we on occasion able to fully complete this task?
As long as we happen to know about an anti-derivative of the integrand, FOTC guarantees us that our impossible summation effort can be completed in a trivial manner.

There exists, however, no fool-proof technique of constructing anti-derivatives other than by calculating zillions of definite integrals!

Thus, it should come as no surprise that it is only in special cases we may find a nice expression for an anti-derivative, or be able to compute some particular definite integral exactly.

 

[saltydog]

Arildno, let me first say, I yield to you sir. With that said, I hold there is a fool-proof technique of constructing the anti-derivative F(x), of any continuous function f(x) no matter how complex and this is guaranteed by the fundamental theorem of Calculus. It is:

$$F(x)=\int_0^x f(t) dt$$

How wonderful it is our world is so complex for only such a world would give rise to us.

 

[arildno]

As I said:

There exists, however, no fool-proof technique of constructing anti-derivatives other than by calculating zillions of definite integrals!

The set of function values to your anti-derivative cannot, in general, be computed in any other way than through calculating zillions of definite integrals (barring the special case where you recognize f to be the derivative of some known function F).

 

[HallsofIvy]

Warning for people who might think they are serious:
saltydog and Arildno are using different meaning for "construct the anti-derivative"!

 

[Izzhov]

Hmm... if I wanted to see a graph of the integral of x^x, would I just have to make it myself (by finding "zillions of definite integrals"), or is there any software that can graph it for me?

 

[mathwonk]

let F(x) = the area under the graph of y = x^x, between 1 and x, say for x>0. then F'(x) = x^x.

where the area is defined of course by the limit of riemann sums.

if you waNT A FORMULA, you coulod write out the powers eries for e^[ ], and subtitute to get the powers eries for x^x = e^[xln(x)], and then antidifferentiate term by term to get F(x).

as pointed out before, no one to my knowledge haS YET GIVEN A NAME TO THIS FUNCTION, SO WE CANNOT SAY ITS NAME, if that is what you mean by tell what it is, BUT WE CAN DEFINE IT AS ABOVE BY A LIMIT, AND BY A SERIES, AND that should do.

If you insist it have a name I suggest calling it Howard, or perhaps Latiph.