Answer
$$\frac{2}{3(x+2)^{3}}-\frac{1}{2(x+2)^{2}}+C$$
Work Step by Step
Given
$$ \int \frac{x d x}{(x+2)^{4}}$$
Let
$$u=x+2 \ \ \ \ \ \to \ \ \ \ du= dx $$
Then
\begin{aligned}
\int \frac{x}{(x+2)^{4}} d x &=\int \frac{u-2}{u^{4}} d u\\
&=\int \frac{u}{u^{4}} d u-\int \frac{2}{u^{4}} d u \\
&=\int u^{-3} d u-\int 2 u^{-4} d u\\
&=\frac{2}{3 u^{3}}-\frac{1}{2 u^{2}}+C \\
&=\frac{2}{3(x+2)^{3}}-\frac{1}{2(x+2)^{2}}+C
\end{aligned}
