Integral de $$$\frac{4}{y}$$$

Encontre $$$\int \frac{4}{y}\, dy$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ usando $$$c=4$$$ e $$$f{\left(y \right)} = \frac{1}{y}$$$:

$${\color{red}{\int{\frac{4}{y} d y}}} = {\color{red}{\left(4 \int{\frac{1}{y} d y}\right)}}$$

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

$$4 {\color{red}{\int{\frac{1}{y} d y}}} = 4 {\color{red}{\ln{\left(\left|{y}\right| \right)}}}$$

Portanto,

$$\int{\frac{4}{y} d y} = 4 \ln{\left(\left|{y}\right| \right)}$$

Adicione a constante de integração:

$$\int{\frac{4}{y} d y} = 4 \ln{\left(\left|{y}\right| \right)}+C$$

$$$\int \frac{4}{y}\, dy = 4 \ln\left(\left|{y}\right|\right) + C$$$A