Integral von $$$\frac{3}{t}$$$

Bestimme $$$\int \frac{3}{t}\, dt$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ mit $$$c=3$$$ und $$$f{\left(t \right)} = \frac{1}{t}$$$ an:

$${\color{red}{\int{\frac{3}{t} d t}}} = {\color{red}{\left(3 \int{\frac{1}{t} d t}\right)}}$$

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

$$3 {\color{red}{\int{\frac{1}{t} d t}}} = 3 {\color{red}{\ln{\left(\left|{t}\right| \right)}}}$$

Daher,

$$\int{\frac{3}{t} d t} = 3 \ln{\left(\left|{t}\right| \right)}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{3}{t} d t} = 3 \ln{\left(\left|{t}\right| \right)}+C$$

$$$\int \frac{3}{t}\, dt = 3 \ln\left(\left|{t}\right|\right) + C$$$A