Integral von $$$\frac{n}{d}$$$ nach $$$d$$$

Bestimme $$$\int \frac{n}{d}\, dd$$$.

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

$${\color{red}{\int{\frac{n}{d} d d}}} = {\color{red}{n \int{\frac{1}{d} d d}}}$$

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

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

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

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

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