Integralen av $$$\frac{\left(x - 2\right) \left(x - 1\right)}{x}$$$

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

Expand the expression:

$${\color{red}{\int{\frac{\left(x - 2\right) \left(x - 1\right)}{x} d x}}} = {\color{red}{\int{\left(x - 3 + \frac{2}{x}\right)d x}}}$$

Integrera termvis:

$${\color{red}{\int{\left(x - 3 + \frac{2}{x}\right)d x}}} = {\color{red}{\left(- \int{3 d x} + \int{\frac{2}{x} d x} + \int{x d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=3$$$:

$$\int{\frac{2}{x} d x} + \int{x d x} - {\color{red}{\int{3 d x}}} = \int{\frac{2}{x} d x} + \int{x d x} - {\color{red}{\left(3 x\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=1$$$:

$$- 3 x + \int{\frac{2}{x} d x} + {\color{red}{\int{x d x}}}=- 3 x + \int{\frac{2}{x} d x} + {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- 3 x + \int{\frac{2}{x} d x} + {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=2$$$ och $$$f{\left(x \right)} = \frac{1}{x}$$$:

$$\frac{x^{2}}{2} - 3 x + {\color{red}{\int{\frac{2}{x} d x}}} = \frac{x^{2}}{2} - 3 x + {\color{red}{\left(2 \int{\frac{1}{x} d x}\right)}}$$

Integralen av $$$\frac{1}{x}$$$ är $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:

$$\frac{x^{2}}{2} - 3 x + 2 {\color{red}{\int{\frac{1}{x} d x}}} = \frac{x^{2}}{2} - 3 x + 2 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

Alltså,

$$\int{\frac{\left(x - 2\right) \left(x - 1\right)}{x} d x} = \frac{x^{2}}{2} - 3 x + 2 \ln{\left(\left|{x}\right| \right)}$$

Lägg till integrationskonstanten:

$$\int{\frac{\left(x - 2\right) \left(x - 1\right)}{x} d x} = \frac{x^{2}}{2} - 3 x + 2 \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \frac{\left(x - 2\right) \left(x - 1\right)}{x}\, dx = \left(\frac{x^{2}}{2} - 3 x + 2 \ln\left(\left|{x}\right|\right)\right) + C$$$A