Integral of $$$- v^{4} + v$$$

Find $$$\int \left(- v^{4} + v\right)\, dv$$$.

Integrate term by term:

$${\color{red}{\int{\left(- v^{4} + v\right)d v}}} = {\color{red}{\left(\int{v d v} - \int{v^{4} d v}\right)}}$$

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

$$- \int{v^{4} d v} + {\color{red}{\int{v d v}}}=- \int{v^{4} d v} + {\color{red}{\frac{v^{1 + 1}}{1 + 1}}}=- \int{v^{4} d v} + {\color{red}{\left(\frac{v^{2}}{2}\right)}}$$

Apply the power rule $$$\int v^{n}\, dv = \frac{v^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=4$$$:

$$\frac{v^{2}}{2} - {\color{red}{\int{v^{4} d v}}}=\frac{v^{2}}{2} - {\color{red}{\frac{v^{1 + 4}}{1 + 4}}}=\frac{v^{2}}{2} - {\color{red}{\left(\frac{v^{5}}{5}\right)}}$$

Therefore,

$$\int{\left(- v^{4} + v\right)d v} = - \frac{v^{5}}{5} + \frac{v^{2}}{2}$$

Add the constant of integration:

$$\int{\left(- v^{4} + v\right)d v} = - \frac{v^{5}}{5} + \frac{v^{2}}{2}+C$$

$$$\int \left(- v^{4} + v\right)\, dv = \left(- \frac{v^{5}}{5} + \frac{v^{2}}{2}\right) + C$$$A