Integraal van $$$\frac{3}{x^{6}}$$$

Bepaal $$$\int \frac{3}{x^{6}}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=3$$$ en $$$f{\left(x \right)} = \frac{1}{x^{6}}$$$:

$${\color{red}{\int{\frac{3}{x^{6}} d x}}} = {\color{red}{\left(3 \int{\frac{1}{x^{6}} d x}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=-6$$$:

$$3 {\color{red}{\int{\frac{1}{x^{6}} d x}}}=3 {\color{red}{\int{x^{-6} d x}}}=3 {\color{red}{\frac{x^{-6 + 1}}{-6 + 1}}}=3 {\color{red}{\left(- \frac{x^{-5}}{5}\right)}}=3 {\color{red}{\left(- \frac{1}{5 x^{5}}\right)}}$$

Dus,

$$\int{\frac{3}{x^{6}} d x} = - \frac{3}{5 x^{5}}$$

Voeg de integratieconstante toe:

$$\int{\frac{3}{x^{6}} d x} = - \frac{3}{5 x^{5}}+C$$

$$$\int \frac{3}{x^{6}}\, dx = - \frac{3}{5 x^{5}} + C$$$A