Integraal van $$$\frac{2}{y}$$$

Bepaal $$$\int \frac{2}{y}\, dy$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ toe met $$$c=2$$$ en $$$f{\left(y \right)} = \frac{1}{y}$$$:

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

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

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

Dus,

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

Voeg de integratieconstante toe:

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

$$$\int \frac{2}{y}\, dy = 2 \ln\left(\left|{y}\right|\right) + C$$$A