Integraal van $$$\frac{z}{\zeta}$$$ met betrekking tot $$$z$$$

Bepaal $$$\int \frac{z}{\zeta}\, dz$$$.

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

$${\color{red}{\int{\frac{z}{\zeta} d z}}} = {\color{red}{\frac{\int{z d z}}{\zeta}}}$$

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

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

Dus,

$$\int{\frac{z}{\zeta} d z} = \frac{z^{2}}{2 \zeta}$$

Voeg de integratieconstante toe:

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

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