Integraal van $$$\frac{3}{2 y}$$$

Bepaal $$$\int \frac{3}{2 y}\, dy$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ toe met $$$c=\frac{3}{2}$$$ en $$$f{\left(y \right)} = \frac{1}{y}$$$:

$${\color{red}{\int{\frac{3}{2 y} d y}}} = {\color{red}{\left(\frac{3 \int{\frac{1}{y} d y}}{2}\right)}}$$

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

$$\frac{3 {\color{red}{\int{\frac{1}{y} d y}}}}{2} = \frac{3 {\color{red}{\ln{\left(\left|{y}\right| \right)}}}}{2}$$

Dus,

$$\int{\frac{3}{2 y} d y} = \frac{3 \ln{\left(\left|{y}\right| \right)}}{2}$$

Voeg de integratieconstante toe:

$$\int{\frac{3}{2 y} d y} = \frac{3 \ln{\left(\left|{y}\right| \right)}}{2}+C$$

$$$\int \frac{3}{2 y}\, dy = \frac{3 \ln\left(\left|{y}\right|\right)}{2} + C$$$A