Integralen av $$$- \frac{31 x}{10}$$$

Bestäm $$$\int \left(- \frac{31 x}{10}\right)\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=- \frac{31}{10}$$$ och $$$f{\left(x \right)} = x$$$:

$${\color{red}{\int{\left(- \frac{31 x}{10}\right)d x}}} = {\color{red}{\left(- \frac{31 \int{x d x}}{10}\right)}}$$

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

$$- \frac{31 {\color{red}{\int{x d x}}}}{10}=- \frac{31 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{10}=- \frac{31 {\color{red}{\left(\frac{x^{2}}{2}\right)}}}{10}$$

Alltså,

$$\int{\left(- \frac{31 x}{10}\right)d x} = - \frac{31 x^{2}}{20}$$

Lägg till integrationskonstanten:

$$\int{\left(- \frac{31 x}{10}\right)d x} = - \frac{31 x^{2}}{20}+C$$

$$$\int \left(- \frac{31 x}{10}\right)\, dx = - \frac{31 x^{2}}{20} + C$$$A