105北一女中二招

#3
試求\(\displaystyle \lim_{n\to \infty}\frac{1}{n}\cdot \left(\frac{\pi}{12n}+\frac{3\pi}{12n}+\frac{5\pi}{12n}+\ldots+\frac{(2n-1)\pi}{12n}\right)=\)
[解答]
原式=∫ {0 to 2}  sin(π*x/12) dx - ∫ {0 to 1}  sin(π*x/6) dx

答案: (6-3√3) / π