integral with nice result

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July 10th 2007, 09:00 AM
galactus

integral with nice result

If anyone wishes to tackle it, go ahead and integrate and see what a cool result you get. I thought it was anyway.

$\int_{0}^{1}\frac{x^{4}(1-x)^{4}}{1+x^{2}}dx$

It shouldn't be too bad. Some division and term by term integration.
July 10th 2007, 09:47 AM
ThePerfectHacker

Quote:

Originally Posted by galactus

If anyone wishes to tackle it, go ahead and integrate and see what a cool result you get. I thought it was anyway.

$\int_{0}^{1}\frac{x^{4}(1-x)^{4}}{1+x^{2}}dx$

It shouldn't be too bad. Some division and term by term integration.

Putnam problem.

The result is,
$\frac{22}{7} - \pi$

I seen this like 5 times already. :eek:
July 10th 2007, 10:00 AM
galactus

OK, sorry, I didn't know. I didn't realize it was a Putnam problem. Rather cliche then.
July 10th 2007, 10:04 AM
Jhevon

Quote:

Originally Posted by galactus

OK, sorry, I didn't know. I didn't realize it was a Putnam problem. Rather cliche then.

yes. in fact, it was mentioned on this forum already, see page 2 here. it was used to prove that 22/7 exceeds pi

but thanks for reminding me about that impressive result. i dont get to see it as often as TPH does :D
July 10th 2007, 10:06 AM
ThePerfectHacker

Read about it here.

Here is a much more nastier one I found on AoPs:
$\frac{355}{113}-\pi = \int_0^1\frac{(25+816x^2)[x(1-x)]^8}{3164(1+x^2)} dx$
July 10th 2007, 10:18 AM
galactus

I seen it in "Gamma, exploring Euler's constant" by Julian Havil.

A nice book.
July 10th 2007, 03:14 PM
ThePerfectHacker

Quote:

Originally Posted by galactus

I seen it in "Gamma, exploring Euler's constant" by Julian Havil.

I really want that book! But I cannot buy it since I am reading math textbooks, I do not have time. :( It makes me :mad: angry that I cannot do everything at the same time!

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