TuringTest 3 years ago Definite integral$\int_0^1\frac x{x^3+1}dx$

1. TuringTest

I've had this one on my head for a couple days now, so whoever can help me gets mad respect :)

2. experimentX

you can do that using partial fraction. but this is not quite interesting.

3. TuringTest

the solution from PF I got seemed drastically different than wolf's elegant answer, though I could have just been unable to manipulate the algebra.

4. TuringTest

Is PF really the only way I guess is my real question

5. TuringTest

it seems like there should be a trick

6. experimentX

yeah ... very short and elegant method.

7. TuringTest

I know, that is why I want something besides hacking away with PF which gives http://www.wolframalpha.com/input/?i=integral+x%2F%281%2Bx%5E3%29

8. TuringTest

okay now I see they are the same, so either that was a dumb question, or there is a trick I want to know

9. experimentX

there might be.

10. experimentX

trig subs might work. but this seems more nasty.

11. anonymous

Yeah, PF would definitely work. Probably won't be able to do much, but I'll try to poke around and see if I can find a nicer method.

12. anonymous

Muhahaha

13. amistre64

... id have to brute away at it :/

14. anonymous

It turns out after researching the integrand for a bit that it has an $$extremely$$ nice McLaurin series. I don't have the energy to type out the entire derivation, but you can check it's: $\frac{x}{x^3+1}=x-x^4+x^7-x^{10}+...=x\sum_{n=0}^\infty(-1)^nx^{3n}$ Throwing in this in the integral and doing a partial integration we get: $\int_0^1{\frac{x}{x^3+1}\, dx}=\int_0^1{x\sum_{n=0}^\infty(-1)^nx^{3n}\,dx} \\ =\sum_{n=0}^\infty \frac{(-1)^n}{3n+1}-\sum_{n=0}^\infty(-1)^n\int_0^1{x^{3n}\,dx} \\ =\sum_{n=0}^\infty\frac{(-1)^n}{3n+1}\left(1-\frac{1}{3n+2}\right) \\ =\sum_{n=0}^\infty\frac{(-1)^n}{3n+2}$ Now that sum is a bit harder to calculate, it involves zeta-functions in combination with some subtitutions, but at least it comes out the same as wolf's answer :p

15. anonymous

I think it's a more elegant solution, despite it involving some pretty high level stuff.

16. anonymous

That was fun, when I first saw that MacLaurin series I got some serious chills down my spine.

17. anonymous

Oops, typo, the integral term on second line should be: $-\sum_{n=0}^\infty{\frac{(-1)^n}{3n+1}}\int_0^1{x^{3n+1}\,dx}$

18. anonymous

heh, was gonna say, you lost an x!

19. anonymous

then i saw you just forgot it

20. experimentX

this seems pretty straight forward $\int_0^1{x\sum_{n=0}^\infty(-1)^nx^{3n}\,dx} = \sum_{n=0}^\infty(-1)^n \int_0^1x^{3n+1}\,dx = \sum_{n=0}^\infty {(-1)^n \over 3n+2}$ I get chills when i see these.

21. anonymous

there's some extra stuff in there...

22. experimentX

I'm not quite well with zeta functions ... looks impossible from Fourier series.

23. anonymous

whatever, if it's actually supposed to be there, I can't figure out why..

24. anonymous

Maybe I forgot something else, did it all on paper and copied it over.

25. anonymous

seems it's just as experimentx did it, what's with the first series and the "(1-"

26. experimentX

still ... i never though of this technique. I haven't been using this technique lately. I'll try to think of this way too in the future. gotta sleep ... nearly 3.30 am here.

27. anonymous

If you've gone McLaurin - then why not go full complex contour integration. The poles are obvious -1, +third-root(-1), -third-root(-1)

28. anonymous

Arent the poles $$-1,\sqrt[3]{-1},-(\sqrt[3]{-1})^2$$?

29. anonymous

Isn't this thing just a trig substitution?

30. anonymous

Never mind I see the x cubed. I can't see how to do this without partial fractions :/ .

31. TuringTest

nice @dape that was definitely more what I was looking for, now this integral is interesting again :)

32. anonymous

Yes @dape these are the poles. Yet I am not certain there is an effective choice of contour so that we can find the needed contribution

33. TuringTest

Contour integral are something I certainly learn, but I would then like to see it that way too.

34. TuringTest

certainly need to*

35. anonymous

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36. TuringTest

Having never taken complex analysis I really don't want to pretend that I understand much about integrating in the complex plane, though of course I understand the concept of writing complex numbers in them and roots of unity... basically.

37. experimentX

@Mikael how to you plan to evaluate that series via contour integrals? Don't you need need to change the series into integral?

38. anonymous