# Why is it difficult to integrate x^x

1. ### latyph

8
I tried doing this but could not,why is it so?

2. ### dextercioby

12,327
It's not difficult.Nobody thought of defining a special function to account for its antiderivative.

Daniel.

3. ### Zurtex

1,123
Because it can't be integrated in terms of elementary functions. Most functions are 'not easy' to integrate in this way.

4. ### HallsofIvy

41,260
Staff Emeritus
Or, to say the same thing that Dextercioby and Zurtex said, in different words, because there is no elementary function whose derivative is xx!

5. ### arildno

11,265
A better question would be:
Why is it simple to integrate f(x)=1?

6. ### abia ubong

70
i tried doin it ,but always get to a place i cant continue.who can help integrate [(x^x)(1+LOG[X])]^2.All help will be appreciated

7. ### arildno

11,265
You won't get any, since an anti-derivative of x^x is inexpressible in terms of elementary functions.

8. ### goldi

22
when i plugged it into the integrator of mathematica it gave it back as same...i dont know why it did not do computation.

9. ### Zurtex

1,123
Because as said many times previously in this thread, it can not be integrated in terms of elementary functions.

10. ### goldi

22
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?

11. ### Zurtex

1,123
There are no special functions defined in general mathematics for the integral.

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

12. ### matt grime

9,395
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?

13. ### goldi

22
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...

14. ### arildno

11,265
Any particular definite integral of x^x can, of course be calculated to an arbitrary degree of accuracy by numerical techniques.

15. ### Jameson

787
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.

Last edited: Jun 1, 2005
16. ### dextercioby

12,327
I think you meant

$$dv=\sec x \ dx$$

I'm not sure though...

Daniel.

17. ### Icebreaker

0
Has it been done before? Any interesting properties?

18. ### Pyrrhus

2,204
Look for Liouville's Principle about integration in finite terms.

19. ### goldi

22
that i would have had tried 100 times...after 1 step i am stuck and there is no way out....

20. ### calculus1967

27
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.

Last edited: Jun 2, 2005