Answer
Diverges
Work Step by Step
Let us consider that $I_n=\int_{2}^{n} \dfrac{x^3}{x^4+1} \ dx$
Suppose $a=x^4+1 \implies da=4x^3 \ dx$
Now, we have: $I_n=\dfrac{1}{4} \int_{17}^{n^4+1} \dfrac{1}{a} \ da=\dfrac{1}{4} \ln (\dfrac{n^4+1}{17})$
Now, $I=\lim\limits_{n \to \infty} I_n=\int_{2}^{\infty} \dfrac{x^3}{x^4+1} \ dx\\=\dfrac{1}{4} \lim\limits_{n \to \infty} \ln (\dfrac{n^4+1}{17})\\=\infty$
Because the logarithm and $n$ both are unbounded. So, the limit does not exist. This means that the given integral diverges.