1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 1/ (x^2n + 1)

  1. Jan 2, 2009 #1
    1. The problem statement, all variables and given/known data

    [tex]\int\frac{1}{x^{2n} + 1}[/tex]

    2. Relevant equations

    De moivre's theorem

    3. The attempt at a solution

    I am sorry moderators. I had posted this problem earlier on PF but i got no replies so i had to do it again.

    I have tried almost every possible substitution. But in vain. So just one way left. Factorize the denominator using de moivre's theorem. I dont know why i am using de moivre but i do think that we may get some ease doing this integral by this method
  2. jcsd
  3. Jan 2, 2009 #2


    User Avatar
    Homework Helper

    Did the question state specifically to use De Moivre's theorem?
    What substitutions did you happen to use?

    If you used De Moivre's theorem where did you reach when you put [itex]x=cos\theta + isin\theta[/itex] ?
  4. Jan 2, 2009 #3


    User Avatar
    Science Advisor
    Homework Helper

    Are you doing the definite integral from -infinity to +infinity? In that case, sure, write down the poles using deMoivre and use the residue theorem.
  5. Jan 2, 2009 #4
    No. Its an indefinite integral. And we haven't been taught with the residue theorem.
  6. Jan 2, 2009 #5
    [tex] x = (-1)^{\frac{1}{2n}} [/tex]

    [tex]\int\prod^{2n-1}_{k=1}\frac{1}{(x - cos\frac{2k\pi + \pi}{2n} + isin\frac{2k\pi + \pi}{2n})}[/tex]
    Last edited: Jan 2, 2009
  7. Jan 3, 2009 #6


    User Avatar
    Gold Member

    Well, you say De-Moivere's?
    If you were to define this integral by I(n).
    Then I(0)=x (upto a constant), do this by induction and by parts.
    That will do the trick.
    I(1)=arctg(x), I(2)=(do it by residue theorem for 1/(1+x^4)) for I(3)=S(1/(x^6+1)), the same.

    You can do this with De-Moivere, but I need to think it over.
  8. Jan 3, 2009 #7


    User Avatar
    Gold Member

    I think you need to do the substituition x=cosh(t) or sinh(t) and use the identity cosh^2(t)-sinh^2(t)=1.
  9. Jan 3, 2009 #8


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    I confess that I would solve this problem with power series. But that's probably a bad idea if you haven't learned methods for manipulating series efficiently, and I'm not sure if I got the answer in the simplest form either.
  10. Jan 3, 2009 #9
    We have been not taught with hyper trigonometric functions nor we have been taught to the residue theorem. How ever i would love to learn that. Any good resource which you could recommend me with? And would it be able for you to elaborate the above quoted post?
  11. Jan 3, 2009 #10
    To hell with the answer.I would love to solve the sum even if the answer come in complex numbers. But you could possibly tell me what do you mean by power series.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook