Integralen av $$$- \frac{2 x}{3}$$$

Bestäm $$$\int \left(- \frac{2 x}{3}\right)\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=- \frac{2}{3}$$$ och $$$f{\left(x \right)} = x$$$:

$${\color{red}{\int{\left(- \frac{2 x}{3}\right)d x}}} = {\color{red}{\left(- \frac{2 \int{x d x}}{3}\right)}}$$

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

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

Alltså,

$$\int{\left(- \frac{2 x}{3}\right)d x} = - \frac{x^{2}}{3}$$

Lägg till integrationskonstanten:

$$\int{\left(- \frac{2 x}{3}\right)d x} = - \frac{x^{2}}{3}+C$$

$$$\int \left(- \frac{2 x}{3}\right)\, dx = - \frac{x^{2}}{3} + C$$$A