Answer
Use the trig substitution:
$$ x+3 =\sqrt{21}\sin u\ \ \ \ \to \ \ \ \ dx= \sqrt{21}\cos udu$$
Work Step by Step
Given $$\begin{equation}\int \frac{x d x}{\sqrt{12-6 x-x^{2}}}\end{equation}$$
Since
\begin{align*}
12-6 x-x^{2}&=-[x^{2}+6x-12]\\
&=-[(x+3)^2-12-9]\\
&=21-(x+3)^2
\end{align*}
Then
\begin{align*}
\int \frac{x d x}{\sqrt{12-6 x-x^{2}}}&=\int \frac{x d x}{\sqrt{21-(x+3)^2}}
\end{align*}
Use the trigonometric substitution:
$$ x+3 =\sqrt{21}\sin u\ \ \ \ \to \ \ \ \ dx= \sqrt{21}\cos udu$$
