Integral von $$$x - 2 - \frac{1}{x}$$$

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

Gliedweise integrieren:

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

Wenden Sie die Konstantenregel $$$\int c\, dx = c x$$$ mit $$$c=2$$$ an:

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

Wenden Sie die Potenzregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=1$$$ an:

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

Das Integral von $$$\frac{1}{x}$$$ ist $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:

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

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

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

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