Integralen av $$$v^{2} - v$$$

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

Integrera termvis:

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

Tillämpa potensregeln $$$\int v^{n}\, dv = \frac{v^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=2$$$:

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

Tillämpa potensregeln $$$\int v^{n}\, dv = \frac{v^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=1$$$:

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

Alltså,

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

Förenkla:

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

Lägg till integrationskonstanten:

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

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