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

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

Integrera termvis:

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

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=2$$$:

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

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

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

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

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

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

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

Alltså,

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

Förenkla:

$$\int{\left(x - 2 - \frac{2}{x^{2}}\right)d x} = \frac{x^{2} \left(x - 4\right) + 4}{2 x}$$

Lägg till integrationskonstanten:

$$\int{\left(x - 2 - \frac{2}{x^{2}}\right)d x} = \frac{x^{2} \left(x - 4\right) + 4}{2 x}+C$$

$$$\int \left(x - 2 - \frac{2}{x^{2}}\right)\, dx = \frac{x^{2} \left(x - 4\right) + 4}{2 x} + C$$$A