Integralen av $$$81 x^{16}$$$

Bestäm $$$\int 81 x^{16}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=81$$$ och $$$f{\left(x \right)} = x^{16}$$$:

$${\color{red}{\int{81 x^{16} d x}}} = {\color{red}{\left(81 \int{x^{16} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=16$$$:

$$81 {\color{red}{\int{x^{16} d x}}}=81 {\color{red}{\frac{x^{1 + 16}}{1 + 16}}}=81 {\color{red}{\left(\frac{x^{17}}{17}\right)}}$$

Alltså,

$$\int{81 x^{16} d x} = \frac{81 x^{17}}{17}$$

Lägg till integrationskonstanten:

$$\int{81 x^{16} d x} = \frac{81 x^{17}}{17}+C$$

$$$\int 81 x^{16}\, dx = \frac{81 x^{17}}{17} + C$$$A