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

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

Wende die Konstantenfaktorregel $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$ mit $$$c=\frac{1}{2}$$$ und $$$f{\left(w \right)} = \frac{1}{w}$$$ an:

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

Das Integral von $$$\frac{1}{w}$$$ ist $$$\int{\frac{1}{w} d w} = \ln{\left(\left|{w}\right| \right)}$$$:

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

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

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

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