Answer
converges.
Work Step by Step
Given $$\int_{0}^{1} \frac{d x}{x^{1 / 3}+x^{2 / 3}} $$
Since for $x\geq0$
\begin{align*}
x^{1 / 3} &\leq x^{1 / 3}+x^{2 / 3}\\
\frac{1}{x^{1 / 3}+x^{2 / 3}} &\leq \frac{1}{x^{1 / 3}}
\end{align*}
and
\begin{align*}
\int_{0}^{1}\frac{1}{x^{1 / 3}}dx&= \frac{3}{2}x^{\frac{2}{3}}\bigg|_{0}^{1}\\
&=\frac{2}{3}
\end{align*}
then by the Comparison Test, $\int_{0}^{1} \frac{d x}{x^{1 / 3}+x^{2 / 3}}$ also converges.
