Answer
Diverges
Work Step by Step
Let us consider that $I_n=\int_{2}^{n} x^3 \sqrt {x^4+1} \ dx$
Suppose $a=x^4+1 \implies da=4x^3 \ dx$
Now, we have: $I_n=\dfrac{1}{4} \sqrt a \ da =\dfrac{1}{6} [(n^4+1)^{3/2} -17^{3/2}]$
Now, $I=\lim\limits_{n \to \infty} I_n=\int_{2}^{\infty} x^3 \sqrt {x^4+1} \ dx \\=\dfrac{1}{6} \lim\limits_{n \to \infty} [(n^4+1)^{3/2} -17^{3/2}] \\=\infty$
Because the expression under the limit is unbounded. So, the limit does not exist. This means that the given integral diverges.