Answer
Diverges
Work Step by Step
Let us consider that $I_n=\int_{n}^{-2} (\dfrac{3}{x^3}+1) \ dx$
After integrating, we have: $I_n=[\dfrac{-3}{2x^2}+x]_n^{-2} =\dfrac{3}{2n^2}-\dfrac{3}{8}-(n+2)$
Now, $I=\lim\limits_{n \to -\infty} I_n=\int_{-\infty}^{-2} (\dfrac{-3}{2x^2}+x) \ dx\\=\lim\limits_{n \to -\infty} [\dfrac{3}{2n^2}-\dfrac{3}{8}-(n+2)]\\=\infty$
This means that the limit does not exist and the given integral diverges.