I saw this one and found a cool solution, whats yours?

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- July 17th 2008, 01:59 PMMathstud28A fun integral
I saw this one and found a cool solution, whats yours?

- July 17th 2008, 02:46 PMPaulRS
Let's start by the typical solution:

Thus:

We also have:

then

Since each of those integrals exists separately:

We have: where the Remainder is (see the Remainder of Taylor's Polynomials)

Since : thus:

Therefore the extra term disappears as n tends to infinity:

Thus: - July 17th 2008, 02:56 PMMathstud28
- July 17th 2008, 03:07 PMThePerfectHacker
- July 18th 2008, 07:34 AMgalactus
I may as well give my two cents. I will use the double integral we're all so fond of. I looked around to see if the Kriz may have done this at some time or another, but I couldn't find anything, so I assume not.

An identity we can use is:

So, we can use our double integral thing like so:

Now switch em':

But,

And,

You know, there is a little something that is bothering me. Please straighten me out if need be.

While ding this I noticed something. Now, assuming the limit above is 0 we are in good shape, but isn't it undefined?. I supose as long as . Maybe I am wrong, but something about that is bothering me an little.

As a check I ran it through my 92 and it said undefined. Yet we should get .

I am just a little iffy about this. I reckon if we want it to be 0 we can. That's what I say.(Worried), yet (Happy) - July 18th 2008, 01:13 PMMathstud28
You are forgetting the powerfulness of actual numbers opposed to limits.

Supose that

Then we have

Now if we had that

Where that would be indeterminate but we have

Now this limit is still a number...a non zero number albeit almost zero so we know that no matter how small it gets it never actually reaches zero...therefore - July 18th 2008, 01:18 PMgalactus
You are so right. I just wanted someones input. I was thinking too much and had a brain cramp, I reckon. Anyway, Pretty cool little problem, mathstud.

It's fun seeing the different ways yo go about it. - July 18th 2008, 01:20 PMMathstud28
- July 18th 2008, 02:35 PMKrizalid
Here.