Example 7. Find $\int x^{2}(2 x-1)^{31} d x$.
Solution. We will gradually get rid of the first factor:

$$
\begin{aligned}
\int x^{2}(2 x-1)^{31} d x & =\left\{\begin{aligned}
x^{2}=u, & d u=2 x d x \\
(2 x-1)^{31} d x=d u, & v=\frac{1}{64}(2 x-1)^{32}
\end{aligned}\right\}= \\
& =\frac{x^{2}}{64}(2 x-1)^{32}-\frac{1}{32} \int x(2 x-1)^{32} d x=
\end{aligned}
$$

$$
\begin{aligned}
& =\left\{\begin{aligned}
x & =u, & d u & =d x \\
(2 x-1)^{32} d x & =d u, & v & =\frac{1}{66}(2 x-1)^{33}
\end{aligned}\right\}= \\
& =\frac{x^{2}}{64}(2 x-1)^{32}-\frac{x}{3266}(2 x-1)^{33}+\frac{1}{3266} \int(2 x-1)^{33} d x= \\
& =\frac{x^{2}}{64}(2 x-1)^{32}-\frac{x}{2112}(2 x-1)^{33}+\frac{(2 x-1)^{34}}{143616}+C
\end{aligned}
$$