Answer
Diverges
Work Step by Step
Let us consider that $I_n=\int_{2}^{n} \dfrac{x^2+1}{2x} \ dx$
Now, we have: $I_n=\int_{2}^{n} (\dfrac{x}{2}+\dfrac{1}{2x})\ dx=\dfrac{n^2-4}{4}+\dfrac{1}{2}\ln |\dfrac{n}{2}|$
Now, $I=\lim\limits_{n \to \infty} I_n=\int_{2}^{\infty} \dfrac{x^2+1}{2x} \ dx\\=\lim\limits_{n \to \infty} (\dfrac{n^2-4}{4}+\dfrac{1}{2}\ln |\dfrac{n}{2}|) \\=\infty$
Because the logarithm term and $n^2$ both are unbounded. So, the limit does not exist. This means that the given integral diverges.