Integralen av $$$\frac{8 \left(x - 1\right)}{x^{2}}$$$

Bestäm $$$\int \frac{8 \left(x - 1\right)}{x^{2}}\, dx$$$.

Inmatningen skrivs om: $$$\int{\frac{8 \left(x - 1\right)}{x^{2}} d x}=\int{\frac{8 x - 8}{x^{2}} d x}$$$.

Expand the expression:

$${\color{red}{\int{\frac{8 x - 8}{x^{2}} d x}}} = {\color{red}{\int{\left(\frac{8}{x} - \frac{8}{x^{2}}\right)d x}}}$$

Integrera termvis:

$${\color{red}{\int{\left(\frac{8}{x} - \frac{8}{x^{2}}\right)d x}}} = {\color{red}{\left(- \int{\frac{8}{x^{2}} d x} + \int{\frac{8}{x} d x}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=8$$$ och $$$f{\left(x \right)} = \frac{1}{x^{2}}$$$:

$$\int{\frac{8}{x} d x} - {\color{red}{\int{\frac{8}{x^{2}} d x}}} = \int{\frac{8}{x} d x} - {\color{red}{\left(8 \int{\frac{1}{x^{2}} d x}\right)}}$$

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

$$\int{\frac{8}{x} d x} - 8 {\color{red}{\int{\frac{1}{x^{2}} d x}}}=\int{\frac{8}{x} d x} - 8 {\color{red}{\int{x^{-2} d x}}}=\int{\frac{8}{x} d x} - 8 {\color{red}{\frac{x^{-2 + 1}}{-2 + 1}}}=\int{\frac{8}{x} d x} - 8 {\color{red}{\left(- x^{-1}\right)}}=\int{\frac{8}{x} d x} - 8 {\color{red}{\left(- \frac{1}{x}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=8$$$ och $$$f{\left(x \right)} = \frac{1}{x}$$$:

$${\color{red}{\int{\frac{8}{x} d x}}} + \frac{8}{x} = {\color{red}{\left(8 \int{\frac{1}{x} d x}\right)}} + \frac{8}{x}$$

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

$$8 {\color{red}{\int{\frac{1}{x} d x}}} + \frac{8}{x} = 8 {\color{red}{\ln{\left(\left|{x}\right| \right)}}} + \frac{8}{x}$$

Alltså,

$$\int{\frac{8 x - 8}{x^{2}} d x} = 8 \ln{\left(\left|{x}\right| \right)} + \frac{8}{x}$$

Lägg till integrationskonstanten:

$$\int{\frac{8 x - 8}{x^{2}} d x} = 8 \ln{\left(\left|{x}\right| \right)} + \frac{8}{x}+C$$

$$$\int \frac{8 \left(x - 1\right)}{x^{2}}\, dx = \left(8 \ln\left(\left|{x}\right|\right) + \frac{8}{x}\right) + C$$$A