Integral von $$$\frac{e_{1}}{t}$$$ nach $$$t$$$

Bestimme $$$\int \frac{e_{1}}{t}\, dt$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ mit $$$c=e_{1}$$$ und $$$f{\left(t \right)} = \frac{1}{t}$$$ an:

$${\color{red}{\int{\frac{e_{1}}{t} d t}}} = {\color{red}{e_{1} \int{\frac{1}{t} d t}}}$$

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

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

Daher,

$$\int{\frac{e_{1}}{t} d t} = e_{1} \ln{\left(\left|{t}\right| \right)}$$

Fügen Sie die Integrationskonstante hinzu:

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

$$$\int \frac{e_{1}}{t}\, dt = e_{1} \ln\left(\left|{t}\right|\right) + C$$$A