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

Bestäm $$$\int \frac{y}{e^{8}}\, dy$$$.

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

$${\color{red}{\int{\frac{y}{e^{8}} d y}}} = {\color{red}{\frac{\int{y d y}}{e^{8}}}}$$

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

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

Alltså,

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

Lägg till integrationskonstanten:

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

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