Integral von $$$\frac{m}{d f}$$$ nach $$$d$$$

Bestimme $$$\int \frac{m}{d f}\, dd$$$.

Wende die Konstantenfaktorregel $$$\int c f{\left(d \right)}\, dd = c \int f{\left(d \right)}\, dd$$$ mit $$$c=\frac{m}{f}$$$ und $$$f{\left(d \right)} = \frac{1}{d}$$$ an:

$${\color{red}{\int{\frac{m}{d f} d d}}} = {\color{red}{\frac{m \int{\frac{1}{d} d d}}{f}}}$$

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

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

Daher,

$$\int{\frac{m}{d f} d d} = \frac{m \ln{\left(\left|{d}\right| \right)}}{f}$$

Fügen Sie die Integrationskonstante hinzu:

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

$$$\int \frac{m}{d f}\, dd = \frac{m \ln\left(\left|{d}\right|\right)}{f} + C$$$A