Integralen av $$$\frac{1}{x^{2} - 78 x}$$$

Bestäm $$$\int \frac{1}{x^{2} - 78 x}\, dx$$$.

Utför partialbråksuppdelning (stegen kan ses »):

$${\color{red}{\int{\frac{1}{x^{2} - 78 x} d x}}} = {\color{red}{\int{\left(\frac{1}{78 \left(x - 78\right)} - \frac{1}{78 x}\right)d x}}}$$

Integrera termvis:

$${\color{red}{\int{\left(\frac{1}{78 \left(x - 78\right)} - \frac{1}{78 x}\right)d x}}} = {\color{red}{\left(- \int{\frac{1}{78 x} d x} + \int{\frac{1}{78 \left(x - 78\right)} d x}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{78}$$$ och $$$f{\left(x \right)} = \frac{1}{x}$$$:

$$\int{\frac{1}{78 \left(x - 78\right)} d x} - {\color{red}{\int{\frac{1}{78 x} d x}}} = \int{\frac{1}{78 \left(x - 78\right)} d x} - {\color{red}{\left(\frac{\int{\frac{1}{x} d x}}{78}\right)}}$$

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

$$\int{\frac{1}{78 \left(x - 78\right)} d x} - \frac{{\color{red}{\int{\frac{1}{x} d x}}}}{78} = \int{\frac{1}{78 \left(x - 78\right)} d x} - \frac{{\color{red}{\ln{\left(\left|{x}\right| \right)}}}}{78}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{78}$$$ och $$$f{\left(x \right)} = \frac{1}{x - 78}$$$:

$$- \frac{\ln{\left(\left|{x}\right| \right)}}{78} + {\color{red}{\int{\frac{1}{78 \left(x - 78\right)} d x}}} = - \frac{\ln{\left(\left|{x}\right| \right)}}{78} + {\color{red}{\left(\frac{\int{\frac{1}{x - 78} d x}}{78}\right)}}$$

Låt $$$u=x - 78$$$ vara.

Då $$$du=\left(x - 78\right)^{\prime }dx = 1 dx$$$ (stegen kan ses »), och vi har att $$$dx = du$$$.

Alltså,

$$- \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{{\color{red}{\int{\frac{1}{x - 78} d x}}}}{78} = - \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{78}$$

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

$$- \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{78} = - \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{78}$$

Kom ihåg att $$$u=x - 78$$$:

$$- \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{78} = - \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{\ln{\left(\left|{{\color{red}{\left(x - 78\right)}}}\right| \right)}}{78}$$

Alltså,

$$\int{\frac{1}{x^{2} - 78 x} d x} = - \frac{\ln{\left(\left|{x}\right| \right)}}{78} + \frac{\ln{\left(\left|{x - 78}\right| \right)}}{78}$$

Förenkla:

$$\int{\frac{1}{x^{2} - 78 x} d x} = \frac{- \ln{\left(\left|{x}\right| \right)} + \ln{\left(\left|{x - 78}\right| \right)}}{78}$$

Lägg till integrationskonstanten:

$$\int{\frac{1}{x^{2} - 78 x} d x} = \frac{- \ln{\left(\left|{x}\right| \right)} + \ln{\left(\left|{x - 78}\right| \right)}}{78}+C$$

$$$\int \frac{1}{x^{2} - 78 x}\, dx = \frac{- \ln\left(\left|{x}\right|\right) + \ln\left(\left|{x - 78}\right|\right)}{78} + C$$$A