Answer
Converges
Work Step by Step
Given $$\int_{1}^{\infty} \frac{1}{\sqrt{x^{5}+2}} d x $$
Since
\begin{align*}
\sqrt{x^{5}+2} &\geq \sqrt{x^{5}}=x^{5 / 2}\\
\frac{1}{\sqrt{x^{5}+2}}& \leq \frac{1}{x^{5 / 2}}
\end{align*}
and
\begin{align*}
\int_{1}^{\infty}\frac{1}{x^{5 / 2}}dx&=\lim_{R\to\infty} \int_{1}^{R}\frac{1}{x^{5 / 2}}dx\\
&= \lim_{R\to\infty} -\frac{2}{3x^{\frac{3}{2}}} \bigg|_{1}^{R}\\
&= \lim_{R\to\infty} -\frac{2}{3R^{\frac{3}{2}}}+\frac{2}{3}\\
&=\frac{2}{3}
\end{align*}
Converges; thus $\displaystyle\int_{1}^{\infty} \frac{1}{\sqrt{x^{5}+2}} d x $ also converges.
