Integraal van $$$\frac{x - 9}{x}$$$

Bepaal $$$\int \frac{x - 9}{x}\, dx$$$.

Expand the expression:

$${\color{red}{\int{\frac{x - 9}{x} d x}}} = {\color{red}{\int{\left(1 - \frac{9}{x}\right)d x}}}$$

Integreer termgewijs:

$${\color{red}{\int{\left(1 - \frac{9}{x}\right)d x}}} = {\color{red}{\left(\int{1 d x} - \int{\frac{9}{x} d x}\right)}}$$

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=1$$$:

$$- \int{\frac{9}{x} d x} + {\color{red}{\int{1 d x}}} = - \int{\frac{9}{x} d x} + {\color{red}{x}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=9$$$ en $$$f{\left(x \right)} = \frac{1}{x}$$$:

$$x - {\color{red}{\int{\frac{9}{x} d x}}} = x - {\color{red}{\left(9 \int{\frac{1}{x} d x}\right)}}$$

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

$$x - 9 {\color{red}{\int{\frac{1}{x} d x}}} = x - 9 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

Dus,

$$\int{\frac{x - 9}{x} d x} = x - 9 \ln{\left(\left|{x}\right| \right)}$$

Voeg de integratieconstante toe:

$$\int{\frac{x - 9}{x} d x} = x - 9 \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \frac{x - 9}{x}\, dx = \left(x - 9 \ln\left(\left|{x}\right|\right)\right) + C$$$A