Answer
Divergent
Work Step by Step
$$\int_{0}^{\infty }\frac{x_{2}}{\sqrt{1+x^{3}}}dx=\lim_{t\rightarrow \infty}\int_{0}^{t}\frac{x_{2}}{\sqrt{1+x^{3}}}dx$$
$$=\frac{1}{3}\lim_{t\rightarrow \infty}\int_{0}^{t}\frac{d(1+x^{3})}{\sqrt{1+x^{3}}}$$
$$=\lim_{t\rightarrow \infty}\left| \frac{2}{3}\sqrt{1+x^{3}} \right |_{0}^{t}$$
$$= \lim_{t\rightarrow -\infty} \frac{2}{3}(\sqrt{1+t^{3}}-1)$$
$$=\infty-\frac{2}{3}\ \ ,divergent$$
