Answer
Converges
Work Step by Step
We have: $\int_{1}^{\infty} \dfrac{d x}{1+x^{3}} $
Since $ 0 \leq \dfrac{1}{x^{3}+1} \leq \dfrac{1}{x^{3}}$; for $1\leq x \leq \infty$
Now, $\int_{1}^{\infty} \dfrac{d x}{x^{3}} =[\dfrac{-1}{2x^{2}} ]_{1}^{\infty} \\=\dfrac{1}{2}$
So, the integral $\int_{1}^{\infty} \dfrac{d x}{1+x^{3}} $ converges by the direct comparison test.