Answer
$\frac{1}{117} (3x+8)^{13}-\frac{2}{27} (3x+8)^{12}+C $
Work Step by Step
Given $$\int x(3x+8)^{11}d x$$
Let
$$u=3x+8 \ \ \ \Rightarrow \ \ du=3dx $$
then
\begin{align*}
\int x(3 x+8)^{11} d x &=\frac{1}{3} \int \frac{1}{3}(u-8) u^{11} d u \\ &=\frac{1}{9}\left(u^{12}-8 u^{11}\right) d u \\ &=\frac{1}{9}\left(\frac{1}{13} u^{13}-\frac{8}{12} u^{12}+C\right) \\ &=\frac{1}{117} u^{13}-\frac{2}{27} u^{12}+C \\ &
=\frac{1}{117} (3x+8)^{13}-\frac{2}{27} (3x+8)^{12}+C
\end{align*}
