Integralen av $$$x^{2} - x - \frac{2}{x - 2}$$$

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

Integrera termvis:

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

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=2$$$:

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

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

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

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

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

Låt $$$u=x - 2$$$ vara.

Då $$$du=\left(x - 2\right)^{\prime }dx = 1 dx$$$ (stegen kan ses »), och vi har att $$$dx = du$$$.

Alltså,

$$\frac{x^{3}}{3} - \frac{x^{2}}{2} - 2 {\color{red}{\int{\frac{1}{x - 2} d x}}} = \frac{x^{3}}{3} - \frac{x^{2}}{2} - 2 {\color{red}{\int{\frac{1}{u} d u}}}$$

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

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

Kom ihåg att $$$u=x - 2$$$:

$$\frac{x^{3}}{3} - \frac{x^{2}}{2} - 2 \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = \frac{x^{3}}{3} - \frac{x^{2}}{2} - 2 \ln{\left(\left|{{\color{red}{\left(x - 2\right)}}}\right| \right)}$$

Alltså,

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

Lägg till integrationskonstanten:

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

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