Integral dari $$$\frac{1}{x^{2} - x}$$$

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

Lakukan dekomposisi pecahan parsial (langkah-langkah dapat dilihat di »):

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

Integralkan suku demi suku:

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

Misalkan $$$u=x - 1$$$.

Kemudian $$$du=\left(x - 1\right)^{\prime }dx = 1 dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$dx = du$$$.

Integral tersebut dapat ditulis ulang sebagai

$$- \int{\frac{1}{x} d x} + {\color{red}{\int{\frac{1}{x - 1} d x}}} = - \int{\frac{1}{x} d x} + {\color{red}{\int{\frac{1}{u} d u}}}$$

Integral dari $$$\frac{1}{u}$$$ adalah $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

Ingat bahwa $$$u=x - 1$$$:

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

Integral dari $$$\frac{1}{x}$$$ adalah $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:

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

Oleh karena itu,

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

Tambahkan konstanta integrasi:

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

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