Integraal van $$$\frac{d}{t}$$$ met betrekking tot $$$t$$$

Bepaal $$$\int \frac{d}{t}\, dt$$$.

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

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

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

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

Dus,

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

Voeg de integratieconstante toe:

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

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