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

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

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

$${\color{red}{\int{\frac{a^{2}}{x^{2}} d x}}} = {\color{red}{a^{2} \int{\frac{1}{x^{2}} d x}}}$$

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

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

Dus,

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

Voeg de integratieconstante toe:

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

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