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5.2: Певний інтеграл
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\[\begin{aligned} \int_1^2 x^2~\dx ~&=~ \lim_{n \to \infty}~\sum_{i=1}^{n} f(x_i^*)\,\Delta x_i ~=~ \lim_{n \to \infty}~\sum_{i=1}^{n} f(x_{i-1})\,\frac{1}{n} ~=~ \lim_{n \to \infty}~\sum_{i=1}^{n} x_...\[\begin{aligned} \int_1^2 x^2~\dx ~&=~ \lim_{n \to \infty}~\sum_{i=1}^{n} f(x_i^*)\,\Delta x_i ~=~ \lim_{n \to \infty}~\sum_{i=1}^{n} f(x_{i-1})\,\frac{1}{n} ~=~ \lim_{n \to \infty}~\sum_{i=1}^{n} x_{i-1}^2\,\frac{1}{n}\\[6pt] &=~ \lim_{n \to \infty}~\sum_{i=1}^{n} \left(1 + \frac{i-1}{n}\right)^2\frac{1}{n} ~=~ \lim_{n \to \infty}~\sum_{i=1}^{n} \left(\frac{1}{n} ~+~ \frac{2}{n^2}(i-1) ~+~ \frac{1}{n^3}(i-1)^2\right)\\[6pt] &=~ \lim_{n \to \infty}~\left(\sum_{i=1}^{n}\frac{1}{n} ~+~ \frac{2}{…