Integraal van $$$\frac{1}{256 x^{16}}$$$

Bepaal $$$\int \frac{1}{256 x^{16}}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{1}{256}$$$ en $$$f{\left(x \right)} = \frac{1}{x^{16}}$$$:

$${\color{red}{\int{\frac{1}{256 x^{16}} d x}}} = {\color{red}{\left(\frac{\int{\frac{1}{x^{16}} d x}}{256}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=-16$$$:

$$\frac{{\color{red}{\int{\frac{1}{x^{16}} d x}}}}{256}=\frac{{\color{red}{\int{x^{-16} d x}}}}{256}=\frac{{\color{red}{\frac{x^{-16 + 1}}{-16 + 1}}}}{256}=\frac{{\color{red}{\left(- \frac{x^{-15}}{15}\right)}}}{256}=\frac{{\color{red}{\left(- \frac{1}{15 x^{15}}\right)}}}{256}$$

Dus,

$$\int{\frac{1}{256 x^{16}} d x} = - \frac{1}{3840 x^{15}}$$

Voeg de integratieconstante toe:

$$\int{\frac{1}{256 x^{16}} d x} = - \frac{1}{3840 x^{15}}+C$$

$$$\int \frac{1}{256 x^{16}}\, dx = - \frac{1}{3840 x^{15}} + C$$$A