Integralen av $$$- \frac{x^{21}}{50}$$$

Bestäm $$$\int \left(- \frac{x^{21}}{50}\right)\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=- \frac{1}{50}$$$ och $$$f{\left(x \right)} = x^{21}$$$:

$${\color{red}{\int{\left(- \frac{x^{21}}{50}\right)d x}}} = {\color{red}{\left(- \frac{\int{x^{21} d x}}{50}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=21$$$:

$$- \frac{{\color{red}{\int{x^{21} d x}}}}{50}=- \frac{{\color{red}{\frac{x^{1 + 21}}{1 + 21}}}}{50}=- \frac{{\color{red}{\left(\frac{x^{22}}{22}\right)}}}{50}$$

Alltså,

$$\int{\left(- \frac{x^{21}}{50}\right)d x} = - \frac{x^{22}}{1100}$$

Lägg till integrationskonstanten:

$$\int{\left(- \frac{x^{21}}{50}\right)d x} = - \frac{x^{22}}{1100}+C$$

$$$\int \left(- \frac{x^{21}}{50}\right)\, dx = - \frac{x^{22}}{1100} + C$$$A