Answer
$$ \frac{\pi}{3}$$
Work Step by Step
Given $$\int_{0}^{3 \sqrt{3} / 2} \frac{d x}{\sqrt{9-x^{2}}}$$
Then
\begin{aligned}
\int_{0}^{\frac{3 \sqrt{3}}{2}} \frac{d x}{\sqrt{9-x^{2}}}
&=\int_{0}^{\frac{3 \sqrt{3}}{2}} \frac{d x}{\sqrt{9\left(1-(x / 3)^{2}\right)}} \\
&=\int_{0}^{\frac{3 \sqrt{3}}{2}} \frac{d x}{3 \sqrt{\left(1-(x / 3)^{2}\right)}} \\
&=\left.\sin ^{-1}\left(\frac{x}{3}\right)\right|_{0} ^{\frac{3 \sqrt{3}}{2}}\\
&= \frac{\pi}{3}
\end{aligned}
