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

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

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

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

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

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

Dus,

$$\int{\left(- \frac{x^{2}}{y^{2}}\right)d x} = - \frac{x^{3}}{3 y^{2}}$$

Voeg de integratieconstante toe:

$$\int{\left(- \frac{x^{2}}{y^{2}}\right)d x} = - \frac{x^{3}}{3 y^{2}}+C$$

$$$\int \left(- \frac{x^{2}}{y^{2}}\right)\, dx = - \frac{x^{3}}{3 y^{2}} + C$$$A