一、题目
已知：
$$
\begin{aligned}
I & = \\
& \lim \limits_{n \rightarrow \infty}\left(\frac{\sin ^{2} \frac{\pi}{n}}{n}+\frac{\sin ^{2} \frac{2 \pi}{n}}{n}+\cdots+\frac{\sin ^{2} \frac{n \pi}{n}}{n}\right)
\end{aligned}
$$
则：
$$
I \ = \ ?
$$
难度评级：
二、解析 
$$
\begin{aligned}
I \\ \\
& = \lim \limits_{n \rightarrow \infty}\left(\frac{\sin ^{2} \frac{\pi}{n}}{n}+\frac{\sin ^{2} \frac{2 \pi}{n}}{n}+\cdots+\frac{\sin ^{2} \frac{n \pi}{n}}{n}\right) \\ \\
& = \frac{1}{n} \lim \limits_{n \rightarrow \infty}\left( \sin ^{2} \frac{\pi}{n} + \sin ^{2} \frac{2 \pi}{n} + \cdots + \sin ^{2} \frac{n \pi}{n} \right) \\ \\
& = \lim \limits_{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^{n} \sin ^{2} \left( \pi \cdot \frac{i}{n} \right) \\ \\
& = \int_{0}^{1} \sin ^{2} ( \pi x ) \mathrm{~d} x \\ \\
& = \frac{1}{\pi} \int_{0}^{1} \sin ^{2} \pi x \mathrm{~d}(\pi x) \\ \\
& \xlongequal{\pi x=t} \frac{1}{\pi} \int_{0}^{\pi} \sin ^{2} t \mathrm{~d} t \\ \\
& = \frac{2}{\pi} \int_{0}^{\frac{\pi}{2}} \sin ^{2} t \mathrm{~d} t \\ \\
& = \frac{2}{\pi} \cdot \frac{1}{2} \cdot \frac{\pi}{2} \\ \\
& = \frac{1}{2}
\end{aligned}
$$