Integral von $$$\frac{4}{y}$$$

Bestimme $$$\int \frac{4}{y}\, dy$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ mit $$$c=4$$$ und $$$f{\left(y \right)} = \frac{1}{y}$$$ an:

$${\color{red}{\int{\frac{4}{y} d y}}} = {\color{red}{\left(4 \int{\frac{1}{y} d y}\right)}}$$

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

$$4 {\color{red}{\int{\frac{1}{y} d y}}} = 4 {\color{red}{\ln{\left(\left|{y}\right| \right)}}}$$

Daher,

$$\int{\frac{4}{y} d y} = 4 \ln{\left(\left|{y}\right| \right)}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{4}{y} d y} = 4 \ln{\left(\left|{y}\right| \right)}+C$$

$$$\int \frac{4}{y}\, dy = 4 \ln\left(\left|{y}\right|\right) + C$$$A