Integraal van $$$\frac{2}{x^{3}}$$$

Bepaal $$$\int \frac{2}{x^{3}}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=2$$$ en $$$f{\left(x \right)} = \frac{1}{x^{3}}$$$:

$${\color{red}{\int{\frac{2}{x^{3}} d x}}} = {\color{red}{\left(2 \int{\frac{1}{x^{3}} d x}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=-3$$$:

$$2 {\color{red}{\int{\frac{1}{x^{3}} d x}}}=2 {\color{red}{\int{x^{-3} d x}}}=2 {\color{red}{\frac{x^{-3 + 1}}{-3 + 1}}}=2 {\color{red}{\left(- \frac{x^{-2}}{2}\right)}}=2 {\color{red}{\left(- \frac{1}{2 x^{2}}\right)}}$$

Dus,

$$\int{\frac{2}{x^{3}} d x} = - \frac{1}{x^{2}}$$

Voeg de integratieconstante toe:

$$\int{\frac{2}{x^{3}} d x} = - \frac{1}{x^{2}}+C$$

$$$\int \frac{2}{x^{3}}\, dx = - \frac{1}{x^{2}} + C$$$A