Integraal van $$$\frac{1}{a x}$$$ met betrekking tot $$$x$$$

Bepaal $$$\int \frac{1}{a x}\, dx$$$.

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

$${\color{red}{\int{\frac{1}{a x} d x}}} = {\color{red}{\frac{\int{\frac{1}{x} d x}}{a}}}$$

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

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

Dus,

$$\int{\frac{1}{a x} d x} = \frac{\ln{\left(\left|{x}\right| \right)}}{a}$$

Voeg de integratieconstante toe:

$$\int{\frac{1}{a x} d x} = \frac{\ln{\left(\left|{x}\right| \right)}}{a}+C$$

$$$\int \frac{1}{a x}\, dx = \frac{\ln\left(\left|{x}\right|\right)}{a} + C$$$A