\(\int \dfrac {1}{\sqrt{9 - 4x^2}} \: dx\)
\begin{equation*}
\begin{split}
& \int \frac {1}{\sqrt{9 - 4x^2}} \: dx \\\\
& \int \frac {1}{\sqrt{3^2 - (2x)^2}} \: dx
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
2x & = 3 \sin \theta \rightarrow \sin \theta = \frac {2x}{3} \\\\
2 \: dx & = 3 \cos \theta \: d\theta \\\\
dx & = \frac 32 \cos \theta \: d\theta
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
& \int \frac {1}{\sqrt{9 - 4x^2}} \: dx \\\\
& \int \frac {1}{\sqrt{3^2 - (2x)^2}} \: dx \\\\
& \int \frac {1}{\sqrt{3^2 - (3 \sin \theta)^2}} \:.\: \frac 32 \cos \theta \: d\theta \\\\
& \frac 32 \int \frac {1}{\sqrt{9 - 9 \sin^2 \theta}} \:.\: \cos \theta \: d\theta \\\\
& \frac 32 \int \frac {1}{\sqrt{9 (1 - \sin^2 \theta}} \:.\: \cos \theta \: d\theta \\\\
& \frac 32 \int \frac {1}{3 \sqrt{\cos^2 \theta}} \:.\: \cos \theta \: d\theta \\\\
& \frac 12 \int \frac {1}{\cos \theta} \:.\: \cos \theta \: d\theta \\\\
&\frac 12 \int \: d\theta \\\\
& \frac 12 \theta + C \\\\
& \frac 12 \: \arcsin \left(\frac {2x}{3}\right) + C
\end{split}
\end{equation*}