Integralen av $$$\frac{6}{x^{2} - 22 x}$$$

Bestäm $$$\int \frac{6}{x^{2} - 22 x}\, dx$$$.

Förenkla integranden:

$${\color{red}{\int{\frac{6}{x^{2} - 22 x} d x}}} = {\color{red}{\int{\frac{6}{x \left(x - 22\right)} d x}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=6$$$ och $$$f{\left(x \right)} = \frac{1}{x \left(x - 22\right)}$$$:

$${\color{red}{\int{\frac{6}{x \left(x - 22\right)} d x}}} = {\color{red}{\left(6 \int{\frac{1}{x \left(x - 22\right)} d x}\right)}}$$

Utför partialbråksuppdelning (stegen kan ses »):

$$6 {\color{red}{\int{\frac{1}{x \left(x - 22\right)} d x}}} = 6 {\color{red}{\int{\left(\frac{1}{22 \left(x - 22\right)} - \frac{1}{22 x}\right)d x}}}$$

Integrera termvis:

$$6 {\color{red}{\int{\left(\frac{1}{22 \left(x - 22\right)} - \frac{1}{22 x}\right)d x}}} = 6 {\color{red}{\left(- \int{\frac{1}{22 x} d x} + \int{\frac{1}{22 \left(x - 22\right)} d x}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{22}$$$ och $$$f{\left(x \right)} = \frac{1}{x}$$$:

$$6 \int{\frac{1}{22 \left(x - 22\right)} d x} - 6 {\color{red}{\int{\frac{1}{22 x} d x}}} = 6 \int{\frac{1}{22 \left(x - 22\right)} d x} - 6 {\color{red}{\left(\frac{\int{\frac{1}{x} d x}}{22}\right)}}$$

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

$$6 \int{\frac{1}{22 \left(x - 22\right)} d x} - \frac{3 {\color{red}{\int{\frac{1}{x} d x}}}}{11} = 6 \int{\frac{1}{22 \left(x - 22\right)} d x} - \frac{3 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}}{11}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{22}$$$ och $$$f{\left(x \right)} = \frac{1}{x - 22}$$$:

$$- \frac{3 \ln{\left(\left|{x}\right| \right)}}{11} + 6 {\color{red}{\int{\frac{1}{22 \left(x - 22\right)} d x}}} = - \frac{3 \ln{\left(\left|{x}\right| \right)}}{11} + 6 {\color{red}{\left(\frac{\int{\frac{1}{x - 22} d x}}{22}\right)}}$$

Låt $$$u=x - 22$$$ vara.

Då $$$du=\left(x - 22\right)^{\prime }dx = 1 dx$$$ (stegen kan ses »), och vi har att $$$dx = du$$$.

Alltså,

$$- \frac{3 \ln{\left(\left|{x}\right| \right)}}{11} + \frac{3 {\color{red}{\int{\frac{1}{x - 22} d x}}}}{11} = - \frac{3 \ln{\left(\left|{x}\right| \right)}}{11} + \frac{3 {\color{red}{\int{\frac{1}{u} d u}}}}{11}$$

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

$$- \frac{3 \ln{\left(\left|{x}\right| \right)}}{11} + \frac{3 {\color{red}{\int{\frac{1}{u} d u}}}}{11} = - \frac{3 \ln{\left(\left|{x}\right| \right)}}{11} + \frac{3 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{11}$$

Kom ihåg att $$$u=x - 22$$$:

$$- \frac{3 \ln{\left(\left|{x}\right| \right)}}{11} + \frac{3 \ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{11} = - \frac{3 \ln{\left(\left|{x}\right| \right)}}{11} + \frac{3 \ln{\left(\left|{{\color{red}{\left(x - 22\right)}}}\right| \right)}}{11}$$

Alltså,

$$\int{\frac{6}{x^{2} - 22 x} d x} = - \frac{3 \ln{\left(\left|{x}\right| \right)}}{11} + \frac{3 \ln{\left(\left|{x - 22}\right| \right)}}{11}$$

Förenkla:

$$\int{\frac{6}{x^{2} - 22 x} d x} = \frac{3 \left(- \ln{\left(\left|{x}\right| \right)} + \ln{\left(\left|{x - 22}\right| \right)}\right)}{11}$$

Lägg till integrationskonstanten:

$$\int{\frac{6}{x^{2} - 22 x} d x} = \frac{3 \left(- \ln{\left(\left|{x}\right| \right)} + \ln{\left(\left|{x - 22}\right| \right)}\right)}{11}+C$$

$$$\int \frac{6}{x^{2} - 22 x}\, dx = \frac{3 \left(- \ln\left(\left|{x}\right|\right) + \ln\left(\left|{x - 22}\right|\right)\right)}{11} + C$$$A