Answer
Converges
Work Step by Step
Given $$\int_{1}^{\infty} \frac{d x}{\left(x^{3}+2 x+4\right)^{1 / 2}} $$
Since
\begin{align*}
\left(x^{3}+2 x+4\right)^{1 / 2} &\geq x^{3 / 2}\\
\frac{1}{\left(x^{3}+2 x+4\right)^{1 / 2}}&\leq \frac{1}{x^{3 / 2}}
\end{align*}
and
\begin{align*}
\int_{1}^{\infty}\frac{1}{x^{3 / 2}}dx&=\lim_{R\to\infty} \int_{1}^{R}\frac{1}{x^{3 / 2}}dx\\
&= \lim_{R\to\infty} -\frac{2}{\sqrt{x}} \bigg|_{1}^{R}\\
&= \lim_{R\to\infty} -\frac{2}{\sqrt{R}}+2\\
&=2
\end{align*}
Converges; then $\displaystyle\int_{1}^{\infty} \frac{d x}{\left(x^{3}+2 x+4\right)^{1 / 2}}$ also converges.
