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

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

Integreer termgewijs:

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

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

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

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

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

Dus,

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

Voeg de integratieconstante toe:

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

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