Integral de $$$\frac{1}{2 w}$$$

Encontre $$$\int \frac{1}{2 w}\, dw$$$.

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

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

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

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

Portanto,

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

Adicione a constante de integração:

$$\int{\frac{1}{2 w} d w} = \frac{\ln{\left(\left|{w}\right| \right)}}{2}+C$$

$$$\int \frac{1}{2 w}\, dw = \frac{\ln\left(\left|{w}\right|\right)}{2} + C$$$A