Answer
Converges
Work Step by Step
$$\int_{1}^{\infty} \frac{d x}{1+x^{3}} $$
Since for $1 \leq x\lt\infty$ $$ 0 \leq \frac{1}{x^{3}+1} \leq \frac{1}{x^{3}}$$ and
\begin{align*}\int_{1}^{\infty} \frac{d x}{x^{3}} &= \frac{-1}{2x^{2}}\bigg|_{1}^{\infty} \\
&=\frac{1}{2}
\end{align*}
converges, then $ \int_{1}^{\infty} \frac{d x}{1+x^{3}}$ also converges.