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

Encontre $$$\int \frac{3}{2 x}\, dx$$$.

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

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

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

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

Portanto,

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

Adicione a constante de integração:

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

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