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

Bepaal $$$\int \frac{x}{x - 1}\, dx$$$.

Herschrijf en splits de breuk:

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

Integreer termgewijs:

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

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

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

Zij $$$u=x - 1$$$.

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

Dus,

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

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

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

We herinneren eraan dat $$$u=x - 1$$$:

$$x + \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = x + \ln{\left(\left|{{\color{red}{\left(x - 1\right)}}}\right| \right)}$$

Dus,

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

Voeg de integratieconstante toe:

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

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