Answer
Diverges
Work Step by Step
Let us consider that $I_n=\int_{2}^{n} \dfrac{2x}{x^2+1} \ dx$
Suppose $a=t^2+1 \implies da=2xdx$
Now, we have: $I_n=\int_{5}^{n^2+1} \dfrac{1}{a} \ da=[\ln |t| ]_5^{n^2+1}=\ln (\dfrac{n^2+1}{5})$
Now, $I=\lim\limits_{n \to \infty} I_n=\int_{2}^{\infty} \dfrac{2x}{x^2+1} \ dx\\=\lim\limits_{n \to \infty} \ln (\dfrac{n^2+1}{5}) \\=\infty$
Because the logarithm and $n$ both are unbounded. So, the limit does not exist. This means that the given integral diverges.