Integralen av $$$\frac{a^{2}}{2}$$$

Bestäm $$$\int \frac{a^{2}}{2}\, da$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(a \right)}\, da = c \int f{\left(a \right)}\, da$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(a \right)} = a^{2}$$$:

$${\color{red}{\int{\frac{a^{2}}{2} d a}}} = {\color{red}{\left(\frac{\int{a^{2} d a}}{2}\right)}}$$

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

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

Alltså,

$$\int{\frac{a^{2}}{2} d a} = \frac{a^{3}}{6}$$

Lägg till integrationskonstanten:

$$\int{\frac{a^{2}}{2} d a} = \frac{a^{3}}{6}+C$$

$$$\int \frac{a^{2}}{2}\, da = \frac{a^{3}}{6} + C$$$A