Integraal van $$$\frac{1}{x^{2} - 78 x}$$$

Bepaal $$$\int \frac{1}{x^{2} - 78 x}\, dx$$$.

Voer een ontbinding in partiële breuken uit (stappen zijn te zien »):

$${\color{red}{\int{\frac{1}{x^{2} - 78 x} d x}}} = {\color{red}{\int{\left(\frac{1}{78 \left(x - 78\right)} - \frac{1}{78 x}\right)d x}}}$$

Integreer termgewijs:

$${\color{red}{\int{\left(\frac{1}{78 \left(x - 78\right)} - \frac{1}{78 x}\right)d x}}} = {\color{red}{\left(- \int{\frac{1}{78 x} d x} + \int{\frac{1}{78 \left(x - 78\right)} d x}\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{1}{78}$$$ en $$$f{\left(x \right)} = \frac{1}{x}$$$:

$$\int{\frac{1}{78 \left(x - 78\right)} d x} - {\color{red}{\int{\frac{1}{78 x} d x}}} = \int{\frac{1}{78 \left(x - 78\right)} d x} - {\color{red}{\left(\frac{\int{\frac{1}{x} d x}}{78}\right)}}$$

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

$$\int{\frac{1}{78 \left(x - 78\right)} d x} - \frac{{\color{red}{\int{\frac{1}{x} d x}}}}{78} = \int{\frac{1}{78 \left(x - 78\right)} d x} - \frac{{\color{red}{\ln{\left(\left|{x}\right| \right)}}}}{78}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{1}{78}$$$ en $$$f{\left(x \right)} = \frac{1}{x - 78}$$$:

$$- \frac{\ln{\left(\left|{x}\right| \right)}}{78} + {\color{red}{\int{\frac{1}{78 \left(x - 78\right)} d x}}} = - \frac{\ln{\left(\left|{x}\right| \right)}}{78} + {\color{red}{\left(\frac{\int{\frac{1}{x - 78} d x}}{78}\right)}}$$

Zij $$$u=x - 78$$$.

Dan $$$du=\left(x - 78\right)^{\prime }dx = 1 dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = du$$$.

Dus,

$$- \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{{\color{red}{\int{\frac{1}{x - 78} d x}}}}{78} = - \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{78}$$

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

$$- \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{78} = - \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{78}$$

We herinneren eraan dat $$$u=x - 78$$$:

$$- \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{78} = - \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{\ln{\left(\left|{{\color{red}{\left(x - 78\right)}}}\right| \right)}}{78}$$

Dus,

$$\int{\frac{1}{x^{2} - 78 x} d x} = - \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{\ln{\left(\left|{x - 78}\right| \right)}}{78}$$

Vereenvoudig:

$$\int{\frac{1}{x^{2} - 78 x} d x} = \frac{- \ln{\left(\left|{x}\right| \right)} + \ln{\left(\left|{x - 78}\right| \right)}}{78}$$

Voeg de integratieconstante toe:

$$\int{\frac{1}{x^{2} - 78 x} d x} = \frac{- \ln{\left(\left|{x}\right| \right)} + \ln{\left(\left|{x - 78}\right| \right)}}{78}+C$$

$$$\int \frac{1}{x^{2} - 78 x}\, dx = \frac{- \ln\left(\left|{x}\right|\right) + \ln\left(\left|{x - 78}\right|\right)}{78} + C$$$A