Integraal van $$$-3 - \frac{1}{x}$$$

Bepaal $$$\int \left(-3 - \frac{1}{x}\right)\, dx$$$.

Integreer termgewijs:

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

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=3$$$:

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

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

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

Dus,

$$\int{\left(-3 - \frac{1}{x}\right)d x} = - 3 x - \ln{\left(\left|{x}\right| \right)}$$

Voeg de integratieconstante toe:

$$\int{\left(-3 - \frac{1}{x}\right)d x} = - 3 x - \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \left(-3 - \frac{1}{x}\right)\, dx = \left(- 3 x - \ln\left(\left|{x}\right|\right)\right) + C$$$A