Integralen av $$$\frac{x^{5}}{5}$$$

Bestäm $$$\int \frac{x^{5}}{5}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{5}$$$ och $$$f{\left(x \right)} = x^{5}$$$:

$${\color{red}{\int{\frac{x^{5}}{5} d x}}} = {\color{red}{\left(\frac{\int{x^{5} d x}}{5}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=5$$$:

$$\frac{{\color{red}{\int{x^{5} d x}}}}{5}=\frac{{\color{red}{\frac{x^{1 + 5}}{1 + 5}}}}{5}=\frac{{\color{red}{\left(\frac{x^{6}}{6}\right)}}}{5}$$

Alltså,

$$\int{\frac{x^{5}}{5} d x} = \frac{x^{6}}{30}$$

Lägg till integrationskonstanten:

$$\int{\frac{x^{5}}{5} d x} = \frac{x^{6}}{30}+C$$

$$$\int \frac{x^{5}}{5}\, dx = \frac{x^{6}}{30} + C$$$A