1/ (x^2n + 1)

1. Jan 2, 2009

FedEx

1. The problem statement, all variables and given/known data

$$\int\frac{1}{x^{2n} + 1}$$

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. Jan 2, 2009

rock.freak667

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 $x=cos\theta + isin\theta$ ?

3. Jan 2, 2009

Dick

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.

4. Jan 2, 2009

FedEx

No. Its an indefinite integral. And we haven't been taught with the residue theorem.

5. Jan 2, 2009

FedEx

$$x = (-1)^{\frac{1}{2n}}$$

$$\int\prod^{2n-1}_{k=1}\frac{1}{(x - cos\frac{2k\pi + \pi}{2n} + isin\frac{2k\pi + \pi}{2n})}$$

Last edited: Jan 2, 2009
6. Jan 3, 2009

MathematicalPhysicist

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.

7. Jan 3, 2009

MathematicalPhysicist

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.

8. Jan 3, 2009

Hurkyl

Staff Emeritus
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.

9. Jan 3, 2009

FedEx

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?

10. Jan 3, 2009

FedEx

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.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook