Answer
The integral converges and its value is $\frac{3}{2}3^{2/3}$.
Work Step by Step
We have
$$
\int_{2}^{5} (5-x)^{-1/3} dx=-\int_{2}^{5} (5-x)^{-1/3} d((5-x)\\
=-\frac{1}{2/3}(5-x)^{2/3}|_{2}^{5}=-\frac{3}{2}(0-3^{2/3})=\frac{3}{2}3^{2/3}
$$
Hence, the integral converges and its value is $\frac{3}{2}3^{2/3}$.
