# Thread: integral with nice result

1. ## 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.

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

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

2. 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.

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

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

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

I seen this like 5 times already.

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

4. 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

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

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

A nice book.

7. 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 angry that I cannot do everything at the same time!