35. $\int\left(5 x^{3}-2 x^{2}+3 x-8\right) d x$.

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35. $\int\left(5 x^{3}-2 x^{2}+3 x-8\right) d x$.
Solution. Applying properties 2 and 3, and then formula 1, we get

$$
\begin{aligned}
& \int\left(5 x^{3}-2 x^{2}+3 x-8\right) d x=5 \int x^{3} d x-2 \int x^{2} d x+3 \int x d x-8 \int d x= \\
= & 5 \cdot \frac{x^{4}}{4}-2 \cdot \frac{x^{3}}{3}+3 \cdot \frac{x^{2}}{2}-8 x+C=\frac{5}{4} x^{4}-\frac{2}{3} x^{3}+\frac{3}{2} x^{2}-8 x+C .
\end{aligned}
$$

Here $C$ is the algebraic sum of four arbitrary constant terms, which are part of each integral.

By differentiating this result, we obtain the integrand, i.e., the solution is correct.