Integraal van $$$\frac{4 x}{x - 6}$$$

Bepaal $$$\int \frac{4 x}{x - 6}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=4$$$ en $$$f{\left(x \right)} = \frac{x}{x - 6}$$$:

$${\color{red}{\int{\frac{4 x}{x - 6} d x}}} = {\color{red}{\left(4 \int{\frac{x}{x - 6} d x}\right)}}$$

Herschrijf en splits de breuk:

$$4 {\color{red}{\int{\frac{x}{x - 6} d x}}} = 4 {\color{red}{\int{\left(1 + \frac{6}{x - 6}\right)d x}}}$$

Integreer termgewijs:

$$4 {\color{red}{\int{\left(1 + \frac{6}{x - 6}\right)d x}}} = 4 {\color{red}{\left(\int{1 d x} + \int{\frac{6}{x - 6} d x}\right)}}$$

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

$$4 \int{\frac{6}{x - 6} d x} + 4 {\color{red}{\int{1 d x}}} = 4 \int{\frac{6}{x - 6} d x} + 4 {\color{red}{x}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=6$$$ en $$$f{\left(x \right)} = \frac{1}{x - 6}$$$:

$$4 x + 4 {\color{red}{\int{\frac{6}{x - 6} d x}}} = 4 x + 4 {\color{red}{\left(6 \int{\frac{1}{x - 6} d x}\right)}}$$

Zij $$$u=x - 6$$$.

Dan $$$du=\left(x - 6\right)^{\prime }dx = 1 dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = du$$$.

De integraal wordt

$$4 x + 24 {\color{red}{\int{\frac{1}{x - 6} d x}}} = 4 x + 24 {\color{red}{\int{\frac{1}{u} d u}}}$$

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

$$4 x + 24 {\color{red}{\int{\frac{1}{u} d u}}} = 4 x + 24 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

We herinneren eraan dat $$$u=x - 6$$$:

$$4 x + 24 \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = 4 x + 24 \ln{\left(\left|{{\color{red}{\left(x - 6\right)}}}\right| \right)}$$

Dus,

$$\int{\frac{4 x}{x - 6} d x} = 4 x + 24 \ln{\left(\left|{x - 6}\right| \right)}$$

Voeg de integratieconstante toe:

$$\int{\frac{4 x}{x - 6} d x} = 4 x + 24 \ln{\left(\left|{x - 6}\right| \right)}+C$$

$$$\int \frac{4 x}{x - 6}\, dx = \left(4 x + 24 \ln\left(\left|{x - 6}\right|\right)\right) + C$$$A