Integralen av $$$\frac{x^{31}}{9}$$$

Bestäm $$$\int \frac{x^{31}}{9}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{9}$$$ och $$$f{\left(x \right)} = x^{31}$$$:

$${\color{red}{\int{\frac{x^{31}}{9} d x}}} = {\color{red}{\left(\frac{\int{x^{31} d x}}{9}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=31$$$:

$$\frac{{\color{red}{\int{x^{31} d x}}}}{9}=\frac{{\color{red}{\frac{x^{1 + 31}}{1 + 31}}}}{9}=\frac{{\color{red}{\left(\frac{x^{32}}{32}\right)}}}{9}$$

Alltså,

$$\int{\frac{x^{31}}{9} d x} = \frac{x^{32}}{288}$$

Lägg till integrationskonstanten:

$$\int{\frac{x^{31}}{9} d x} = \frac{x^{32}}{288}+C$$

$$$\int \frac{x^{31}}{9}\, dx = \frac{x^{32}}{288} + C$$$A