To express the integral

$\int_5^8 \left(x^2 + \frac{1}{x}\right) dx$

as a limit of Riemann sums using right endpoints, follow these steps:

General Form of Riemann Sum

The integral can be expressed as: $\int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x,$ where:

• $x_i^*$ is the value of $x$ at the chosen sample point (here, the right endpoint) in the $i$-th subinterval,
• $\Delta x = \frac{b-a}{n}$ is the width of each subinterval,
• $f(x)$ is the integrand.

Setup for This Problem

• $f(x) = x^2 + \frac{1}{x}$,
• $a = 5$,