Integralen av $$$\frac{y}{8}$$$

Bestäm $$$\int \frac{y}{8}\, dy$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ med $$$c=\frac{1}{8}$$$ och $$$f{\left(y \right)} = y$$$:

$${\color{red}{\int{\frac{y}{8} d y}}} = {\color{red}{\left(\frac{\int{y d y}}{8}\right)}}$$

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

$$\frac{{\color{red}{\int{y d y}}}}{8}=\frac{{\color{red}{\frac{y^{1 + 1}}{1 + 1}}}}{8}=\frac{{\color{red}{\left(\frac{y^{2}}{2}\right)}}}{8}$$

Alltså,

$$\int{\frac{y}{8} d y} = \frac{y^{2}}{16}$$

Lägg till integrationskonstanten:

$$\int{\frac{y}{8} d y} = \frac{y^{2}}{16}+C$$

$$$\int \frac{y}{8}\, dy = \frac{y^{2}}{16} + C$$$A