Evaluate: ∫₁³ (x² + x) dx, expressing as a limit of sum.

$\int_a^b f(x)dx = \lim_{n\to\infty} h\sum_{r=0}^{n-1} f(a + rh)$, $h = \frac{b-a}{n} = \frac{2}{n}$, $f(x) = x^2 + x$. $f(1 + rh) = (1+rh)^2 + (1+rh) = 2 + 3rh + r^2h^2$. $\sum_{r=0}^{n-1} f(1+rh) = 2n + 3h\frac{n(n-1)}{2} + h^2\frac{(n-1)n(2n-1)}{6}$. With $h = \frac{2}{n}$: $I = \lim_{n\to\infty}\frac{2}{n}\left[2n + \frac{3(n-1)}{1} + \frac{4(n-1)(2n-1)}{6n}\right]$ $= \lim\left[4 + 6\left(1 - \frac{1}{n}\right) + \frac{4}{3}\left(1 - \frac{1}{n}\right)\left(2 - \frac{1}{n}\right)\right]$ $= 4 + 6 + \frac{8}{3}$. Hence $I = \frac{38}{3}$.