Integraal van $$$\frac{m}{d f}$$$ met betrekking tot $$$d$$$

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

Pas de constante-veelvoudregel $$$\int c f{\left(d \right)}\, dd = c \int f{\left(d \right)}\, dd$$$ toe met $$$c=\frac{m}{f}$$$ en $$$f{\left(d \right)} = \frac{1}{d}$$$:

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

De integraal van $$$\frac{1}{d}$$$ is $$$\int{\frac{1}{d} d d} = \ln{\left(\left|{d}\right| \right)}$$$:

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

Dus,

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

Voeg de integratieconstante toe:

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

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