Integraal van $$$1 + \frac{2}{x}$$$

Bepaal $$$\int \left(1 + \frac{2}{x}\right)\, dx$$$.

Integreer termgewijs:

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

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

$$\int{\frac{2}{x} d x} + {\color{red}{\int{1 d x}}} = \int{\frac{2}{x} d x} + {\color{red}{x}}$$

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

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

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

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

Dus,

$$\int{\left(1 + \frac{2}{x}\right)d x} = x + 2 \ln{\left(\left|{x}\right| \right)}$$

Voeg de integratieconstante toe:

$$\int{\left(1 + \frac{2}{x}\right)d x} = x + 2 \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \left(1 + \frac{2}{x}\right)\, dx = \left(x + 2 \ln\left(\left|{x}\right|\right)\right) + C$$$A