Integral de $$$- \frac{2}{3 x}$$$

Encontre $$$\int \left(- \frac{2}{3 x}\right)\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=- \frac{2}{3}$$$ e $$$f{\left(x \right)} = \frac{1}{x}$$$:

$${\color{red}{\int{\left(- \frac{2}{3 x}\right)d x}}} = {\color{red}{\left(- \frac{2 \int{\frac{1}{x} d x}}{3}\right)}}$$

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

$$- \frac{2 {\color{red}{\int{\frac{1}{x} d x}}}}{3} = - \frac{2 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}}{3}$$

Portanto,

$$\int{\left(- \frac{2}{3 x}\right)d x} = - \frac{2 \ln{\left(\left|{x}\right| \right)}}{3}$$

Adicione a constante de integração:

$$\int{\left(- \frac{2}{3 x}\right)d x} = - \frac{2 \ln{\left(\left|{x}\right| \right)}}{3}+C$$

$$$\int \left(- \frac{2}{3 x}\right)\, dx = - \frac{2 \ln\left(\left|{x}\right|\right)}{3} + C$$$A