$\lim_{n \to \infty} \sum_{k=0}^n \dfrac{k}{n^2 + k^2} = \lim_{n \to \infty} \sum_{k=0}^n \dfrac{1}{n} \dfrac{\dfrac{k}{n}}{1 + \left(\dfrac{k}{n}\right)^2}.$
The right side is a Riemann sum: The interval $[0, 1]$ is divided into $n$ subintervals of width $\dfrac{1}{n}$, and rectangles of height $\dfrac{\dfrac{k}{n}}{1 + \left(\dfrac{k}{n}\right)^2}$ are constructed on the subintervals. The sum is the sum of the areas of the rectangles, an approximation to the area under $f(x) = \dfrac{x}{1 + x^2}$ from $x = 0$ to $x = 1$. The limit lets the number of rectangles go to infinity. This gives the integral
$\int_0^1 \dfrac{x}{1 + x^2}\,dx = \dfrac{1}{2} \ln 2.$