Let be an integer. Evaluate

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- May 14th 2009, 06:40 PMNonCommAlgTechniques of integration (3)
Let be an integer. Evaluate

- May 14th 2009, 07:38 PMpickslides
Using itegration by parts I could probably kick this thing off...

Not really sure how to do,

probably needs some trig identities to tidy it up. - May 14th 2009, 08:22 PMNonCommAlg
- May 15th 2009, 01:10 AMMoo
Hello,

Okay, I think I got the correct idea (Evilgrin)

From , it follows that :

The integral is thus :

Which is :

That's a really nice one (Surprised) - May 15th 2009, 01:17 AMNonCommAlg
Bravo Moo! that's exactly the idea ... although you made a strange mistake at the end of your solution, which i'm sure you'll easily find and fix it! (Wink)

- May 15th 2009, 01:22 AMMoo
- May 17th 2009, 03:03 AMsimplependulum
Consider

Since is real , the terms of sine function will be cancelled finally ,

but we experience that if n-2k is not equal to n so the integral becomes :

- May 17th 2009, 03:23 PMNonCommAlg
- May 17th 2009, 04:05 PMTheEmptySet
It can be done using complex variables...

First note that

Parameterize the unit circle in the comples plane with

So the integral becomes

expanding with the binomial theorem we get

each finite sume is its own larent series so the residues are coeffients on the terms so we get

- May 17th 2009, 04:55 PMNonCommAlg