Integraal van $$$- \frac{y}{x^{2}}$$$ met betrekking tot $$$x$$$

Bepaal $$$\int \left(- \frac{y}{x^{2}}\right)\, dx$$$.

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

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

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=-2$$$:

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

Dus,

$$\int{\left(- \frac{y}{x^{2}}\right)d x} = \frac{y}{x}$$

Voeg de integratieconstante toe:

$$\int{\left(- \frac{y}{x^{2}}\right)d x} = \frac{y}{x}+C$$

$$$\int \left(- \frac{y}{x^{2}}\right)\, dx = \frac{y}{x} + C$$$A