Integraal van $$$\frac{1}{2 w}$$$

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

Pas de constante-veelvoudregel $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$ toe met $$$c=\frac{1}{2}$$$ en $$$f{\left(w \right)} = \frac{1}{w}$$$:

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

De integraal van $$$\frac{1}{w}$$$ is $$$\int{\frac{1}{w} d w} = \ln{\left(\left|{w}\right| \right)}$$$:

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

Dus,

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

Voeg de integratieconstante toe:

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

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