Integral von $$$\frac{2}{35 x}$$$

Bestimme $$$\int \frac{2}{35 x}\, dx$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=\frac{2}{35}$$$ und $$$f{\left(x \right)} = \frac{1}{x}$$$ an:

$${\color{red}{\int{\frac{2}{35 x} d x}}} = {\color{red}{\left(\frac{2 \int{\frac{1}{x} d x}}{35}\right)}}$$

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

$$\frac{2 {\color{red}{\int{\frac{1}{x} d x}}}}{35} = \frac{2 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}}{35}$$

Daher,

$$\int{\frac{2}{35 x} d x} = \frac{2 \ln{\left(\left|{x}\right| \right)}}{35}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{2}{35 x} d x} = \frac{2 \ln{\left(\left|{x}\right| \right)}}{35}+C$$

$$$\int \frac{2}{35 x}\, dx = \frac{2 \ln\left(\left|{x}\right|\right)}{35} + C$$$A