구분구적 극한

$\displaystyle \begin{aligned} \text{준식} ) & = \lim\limits _{n \rightarrow \infty} {} \left\{ \frac{\left( \frac{1}{n} \right) ^{2}}{1 + \left( \frac{1}{n} \right) ^{3}} + \frac{\left( \frac{2}{n} \right) ^{2}}{1 + \left( \frac{2}{n} \right) ^{3}} + \frac{\left( \frac{3}{n} \right) ^{2}}{1 + \left( \frac{3}{n} \right) ^{3}} + \cdots + \frac{\left( \frac{n}{n} \right) ^{2}}{1 + \left( \frac{n}{n} \right) ^{3}} \right\} \times \frac{1}{n} \end{aligned}$ $\displaystyle \begin{aligned} & = \lim\limits _{n \rightarrow \infty} {} \sum\limits _{k = 1} ^{n} \frac{\left( \frac{k}{n} \right) ^{2}}{1 + \left( \frac{k}{n} \right) ^{3}} \times \frac{1}{n} = \int _{0} ^{1} {} \frac{x ^{2}}{1 + x ^{3}} dx \end{aligned}$ $\displaystyle = \frac{1}{3} \left[ \ln \left| 1 + x ^{3} \right| \right] _{0} ^{1} = \frac{1}{3} \ln 2$