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 Post subject: cool integralPosted: Fri, 25 Jul 2008 00:46:01 UTC
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Give this one a go:

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 Post subject: Re: cool integralPosted: Fri, 25 Jul 2008 03:05:49 UTC
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cody wrote:
Give this one a go:

Spoiler:
in the inner integral put then:

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 Post subject: Posted: Fri, 25 Jul 2008 17:57:05 UTC
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Thanks for the response, commutative.

Actually, that is basically how I approached it, except I used u instead of v as a sub. I reckon I won't bother posting my result then. At least now I know I done it correctly.

The original problem was multiplied by -4, therefore, resulting in Pi as the solution.

What I find amazing about these types of integrals is that one can run them through Maple or Mathematica and not get a result( if you do get a result it is a long complicated thing), yet human ingenuity easily pumps out the answer.

I ran this one through Maple and I couldn't get it to give me a solution, rather definite or indefinite.

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 Post subject: Posted: Sat, 26 Jul 2008 06:28:44 UTC
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here's a "cool" integral:

Hint:
Spoiler:
the answer is in terms of .

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 Post subject: Posted: Sat, 26 Jul 2008 19:33:00 UTC
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Hey commutative. That is a cool integral.

I haven't had much time to spend with it, but here are a few quick thoughts I had.

Using parts, I got it to:

Now, this has a series of

Where are the Bernoulli numbers. But what to do with that thing now?.

It would be easier to use

That is as far as I have gotten for now. What is your approach?. I bet it is clever, whatever it is. Parhaps you can show me a few tricks and cool identities to use.

Also, if we make the sub , we get:

This looks nasty, but it may be easier to work with than the others I mentioned.

We can use the identity

Just brainstorming for now.

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 Post subject: Posted: Sun, 27 Jul 2008 13:05:03 UTC
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Well, here is what I managed to do, but I got hung up.

After making the sub

Then, using the aforementioned identity:

But,

Now, using gamma:

But this equals , so I used the beta integral and got:

Now, I got stuck. Perhaps you can help me finish or show me the error of my ways. I am sure there is an easier way, but this looked promising until I got hung up on the last part. Is there some little thing I am not seeing?. Is there a in there somewhere?.

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 Post subject: Posted: Wed, 30 Jul 2008 16:45:52 UTC
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I tried editing my post, but it didn't take.

Here are some other thoughts I had.

We could use the series for ln(1-x).

Since , we can write:

This gives the series:

Another is to use

Then use the series and go from there.

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 Post subject: Posted: Wed, 30 Jul 2008 16:57:36 UTC
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Joined: Sun, 8 Jun 2008 01:27:21 UTC
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Location: Pennsylvania
Sorry for not getting back to you on MHF Galctus...I have been up for 39 hours and I jsut forgot! Sorry...but let me pose this question for Commutative...do you mean to solve this by use of the power series of arcsine squared with reciprocated central binomial coefficients etc? Or is there an easier way?

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 Post subject: Posted: Wed, 30 Jul 2008 18:14:00 UTC
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Whatever way you want to buddy. Using the series and gamma/beta thing would be nice. But whatever. There are many ways to go about it I have found. I get close but not quite. I listed several here and on MHF

WE can even use the series for ln(1-x). Or

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 Post subject: Posted: Tue, 5 Aug 2008 19:46:12 UTC
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Well, it doesn't appear anyone is interested anyway, but I finally found a solution to this infernal thing.

Using

Factor:

Now, if we use the series for ln(1-a), we get

Plus some other stuff to integrate because of the .

This all leads us to the solution:

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