Answer
$$
\int x(x+1)^9 dx= \frac{1}{11}(x+1)^{11}-\frac{1}{10}(x+1)^{10} +c
$$
Work Step by Step
Since $ u=x+1$, then $ du=dx $ and hence, $$
\int x(x+1)^9 dx=\int (u-1)u^9 d u=\int u^{10}-u^9 d u\\
=\frac{1}{11}u^{11}-\frac{1}{10}u^{10} +c=\frac{1}{11}(x+1)^{11}-\frac{1}{10}(x+1)^{10} +c
$$
