Integralen av $$$\frac{1}{v^{3}}$$$

Bestäm $$$\int \frac{1}{v^{3}}\, dv$$$.

Tillämpa potensregeln $$$\int v^{n}\, dv = \frac{v^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=-3$$$:

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

Alltså,

$$\int{\frac{1}{v^{3}} d v} = - \frac{1}{2 v^{2}}$$

Lägg till integrationskonstanten:

$$\int{\frac{1}{v^{3}} d v} = - \frac{1}{2 v^{2}}+C$$

$$$\int \frac{1}{v^{3}}\, dv = - \frac{1}{2 v^{2}} + C$$$A