# A Definite Integral Whose Value Will Be Familiar To Everyone?

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

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