Integralen av $$$\frac{x}{e^{2}}$$$

Bestäm $$$\int \frac{x}{e^{2}}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=e^{-2}$$$ och $$$f{\left(x \right)} = x$$$:

$${\color{red}{\int{\frac{x}{e^{2}} d x}}} = {\color{red}{\frac{\int{x d x}}{e^{2}}}}$$

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

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

Alltså,

$$\int{\frac{x}{e^{2}} d x} = \frac{x^{2}}{2 e^{2}}$$

Lägg till integrationskonstanten:

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

$$$\int \frac{x}{e^{2}}\, dx = \frac{x^{2}}{2 e^{2}} + C$$$A