[再寄小读者之数学篇](2014-06-19 微分等式的结论)

证明: $\dps{\int_0^{2\pi}\sex{\int_x^{2\pi}\cfrac{\sin t}{t}\rd t}\rd x=0}$.  

 

证明: $$\beex \bea \int_0^{2\pi}\sex{\int_x^{2\pi}\cfrac{\sin t}{t}\rd t}\rd x &=\int_0^{2\pi} \int_0^t \cfrac{\sin t}{t}\rd x\rd t\\ &=\int_0^{2\pi} \cfrac{\sin t}{t}\cdot t\rd t\\ &=\int_0^{2\pi}\sin t\rd t\\ &=0. \eea \eeex$$