Integral of $$$- v^{2} - v$$$

Find $$$\int \left(- v^{2} - v\right)\, dv$$$.

Integrate term by term:

$${\color{red}{\int{\left(- v^{2} - v\right)d v}}} = {\color{red}{\left(- \int{v d v} - \int{v^{2} d v}\right)}}$$

Apply the power rule $$$\int v^{n}\, dv = \frac{v^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=1$$$:

$$- \int{v^{2} d v} - {\color{red}{\int{v d v}}}=- \int{v^{2} d v} - {\color{red}{\frac{v^{1 + 1}}{1 + 1}}}=- \int{v^{2} d v} - {\color{red}{\left(\frac{v^{2}}{2}\right)}}$$

Apply the power rule $$$\int v^{n}\, dv = \frac{v^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=2$$$:

$$- \frac{v^{2}}{2} - {\color{red}{\int{v^{2} d v}}}=- \frac{v^{2}}{2} - {\color{red}{\frac{v^{1 + 2}}{1 + 2}}}=- \frac{v^{2}}{2} - {\color{red}{\left(\frac{v^{3}}{3}\right)}}$$

Therefore,

$$\int{\left(- v^{2} - v\right)d v} = - \frac{v^{3}}{3} - \frac{v^{2}}{2}$$

Simplify:

$$\int{\left(- v^{2} - v\right)d v} = \frac{v^{2} \left(- 2 v - 3\right)}{6}$$

Add the constant of integration:

$$\int{\left(- v^{2} - v\right)d v} = \frac{v^{2} \left(- 2 v - 3\right)}{6}+C$$

$$$\int \left(- v^{2} - v\right)\, dv = \frac{v^{2} \left(- 2 v - 3\right)}{6} + C$$$A