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Here's a definite integral whose value carries memories of grade school. Is there a useful generalization ?

$$ \int_0^1 \frac{x^4(1-x)^4}{1+x^2} \ dx = \frac{22}{7} - \pi$$

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Indeed, there is. –  Potato Sep 8 '13 at 21:32
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On 28 November 2003 I created this Wikipedia article, to which various others have contributed since then. I think I was the one who added some generalizations, but I'm not sure.

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(I'm looking for an integral that evaluates to 355/113 - pi) –  Alan Sep 8 '13 at 21:42
    
I'm not sure that would be difficult to do, but if you want it to be something that can be seen to be positive without knowing in advance that $355/113>\pi$, that may be harder. For example, certainly $\displaystyle\int_0^1\left(\frac{355}{113}-\pi\right)\,dx$ $=\dfrac{355}{113}-\pi$, but probably that's not what you're looking for. –  Michael Hardy Sep 8 '13 at 21:45
    
You aren't far off,I thought it would be nice to have an arctangent to evaluate. Reference: "Integral proofs that 355/113 > pi" Stephen K. Lucas –  Alan Sep 8 '13 at 21:56

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