One time I was bored and played around a bit with integrals and wolfram alpha and tested the following integral:
http://www.wolframalpha.com/input/?i=integral_0%5E1+ceil%28x*sin%281%2Fx%29%29
Note: The result at my laptop shows: $ \int_0^1 \lceil { x\sin({1 \over x})} \rceil \,dx = 1 - \frac{\log(4)}{2\pi} $
I was a bit surprised seeing this and wondered, if this result makes any sense and if yes, if there were an explanation, where $\frac{\log(4)}{2\pi}$ exactly comes from. The numbers look too neat to be random, so there might be a way to derive those values directly without using the help of a calculator etc.
As always: Thanks in advance for any constructive answer/comment.