Integral de $$$\frac{1}{2 y}$$$

Encontre $$$\int \frac{1}{2 y}\, dy$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ usando $$$c=\frac{1}{2}$$$ e $$$f{\left(y \right)} = \frac{1}{y}$$$:

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

A integral de $$$\frac{1}{y}$$$ é $$$\int{\frac{1}{y} d y} = \ln{\left(\left|{y}\right| \right)}$$$:

$$\frac{{\color{red}{\int{\frac{1}{y} d y}}}}{2} = \frac{{\color{red}{\ln{\left(\left|{y}\right| \right)}}}}{2}$$

Portanto,

$$\int{\frac{1}{2 y} d y} = \frac{\ln{\left(\left|{y}\right| \right)}}{2}$$

Adicione a constante de integração:

$$\int{\frac{1}{2 y} d y} = \frac{\ln{\left(\left|{y}\right| \right)}}{2}+C$$

$$$\int \frac{1}{2 y}\, dy = \frac{\ln\left(\left|{y}\right|\right)}{2} + C$$$A