Integraal van $$$\frac{7}{z^{2}}$$$

Bepaal $$$\int \frac{7}{z^{2}}\, dz$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(z \right)}\, dz = c \int f{\left(z \right)}\, dz$$$ toe met $$$c=7$$$ en $$$f{\left(z \right)} = \frac{1}{z^{2}}$$$:

$${\color{red}{\int{\frac{7}{z^{2}} d z}}} = {\color{red}{\left(7 \int{\frac{1}{z^{2}} d z}\right)}}$$

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

$$7 {\color{red}{\int{\frac{1}{z^{2}} d z}}}=7 {\color{red}{\int{z^{-2} d z}}}=7 {\color{red}{\frac{z^{-2 + 1}}{-2 + 1}}}=7 {\color{red}{\left(- z^{-1}\right)}}=7 {\color{red}{\left(- \frac{1}{z}\right)}}$$

Dus,

$$\int{\frac{7}{z^{2}} d z} = - \frac{7}{z}$$

Voeg de integratieconstante toe:

$$\int{\frac{7}{z^{2}} d z} = - \frac{7}{z}+C$$

$$$\int \frac{7}{z^{2}}\, dz = - \frac{7}{z} + C$$$A