Answer
Diverges.
Work Step by Step
$$\lim\limits_{R \to 3-}\int_0^R\frac{dx}{(3-x)^{3/2}}\\= \lim\limits_{R \to 3-}\frac{-1}{-1/2}\frac{1}{(3-x)^{1/2}}|_0^R=2\lim\limits_{R \to 3-}(\frac{1}{\sqrt{3-R}}-\frac{1}{\sqrt 3})\\=3(\infty-1)=\infty.$$
Hence, $\int_0^3\frac{dx}{(3-x)^{3/2}}$ diverges.
