32. $\int 8 x^{3} d x$.

The integral of $8x^3$ with respect to $x$ is calculated as follows:

\[
\int 8 x^{3} d x = 8 \int x^{3} d x = 8 \left( \frac{x^{4}}{4} \right) + C = 2x^{4} + C
\]
Solution. Applying property 2 and formula 1, we get

$$
\int 8 x^{3} d x=8 \int x^{3} d x=8 \cdot \frac{x^{4}}{4}+C=2 x^{4}+C
$$

Verification: $d\left(2 x^{4}+C\right)=2 \cdot 4 x^{3} d x=8 x^{3} d x$. The solution is correct.