Integral de $$$\frac{z}{\zeta}$$$ em relação a $$$z$$$

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

Aplique a regra do múltiplo constante $$$\int c f{\left(z \right)}\, dz = c \int f{\left(z \right)}\, dz$$$ usando $$$c=\frac{1}{\zeta}$$$ e $$$f{\left(z \right)} = z$$$:

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

Aplique a regra da potência $$$\int z^{n}\, dz = \frac{z^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=1$$$:

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

Portanto,

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

Adicione a constante de integração:

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

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