Integral de $$$-1 + \frac{1}{y}$$$

Encontre $$$\int \left(-1 + \frac{1}{y}\right)\, dy$$$.

Integre termo a termo:

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

Aplique a regra da constante $$$\int c\, dy = c y$$$ usando $$$c=1$$$:

$$\int{\frac{1}{y} d y} - {\color{red}{\int{1 d y}}} = \int{\frac{1}{y} d y} - {\color{red}{y}}$$

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

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

Portanto,

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

Adicione a constante de integração:

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

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