Answer
$$\frac{2}{3}(x+2)^{3/2} -4\sqrt{x+2}+C$$
Work Step by Step
Given $$\int \frac{x}{\sqrt{x+2}} d x$$
Let $$ u^2=x+2\ \ \ \ \ 2udu=dx$$
Then \begin{align*}
\int \frac{x}{\sqrt{x+2}} d x&=2\int \frac{u(u^2-2)du}{u} \\
&= 2\int (u^2-2)du\\
&=\frac{2}{3}u^3 -4u+C\\
&=\frac{2}{3}(x+2)^{3/2} -4\sqrt{x+2}+C
\end{align*}
