Integral de $$$\frac{\ln\left(2\right)}{x}$$$

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

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

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

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

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

Portanto,

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

Adicione a constante de integração:

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

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