Integral dari $$$\frac{1}{x - 3}$$$

Temukan $$$\int \frac{1}{x - 3}\, dx$$$.

Misalkan $$$u=x - 3$$$.

Kemudian $$$du=\left(x - 3\right)^{\prime }dx = 1 dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$dx = du$$$.

Jadi,

$${\color{red}{\int{\frac{1}{x - 3} d x}}} = {\color{red}{\int{\frac{1}{u} d u}}}$$

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

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

Ingat bahwa $$$u=x - 3$$$:

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

Oleh karena itu,

$$\int{\frac{1}{x - 3} d x} = \ln{\left(\left|{x - 3}\right| \right)}$$

Tambahkan konstanta integrasi:

$$\int{\frac{1}{x - 3} d x} = \ln{\left(\left|{x - 3}\right| \right)}+C$$

$$$\int \frac{1}{x - 3}\, dx = \ln\left(\left|{x - 3}\right|\right) + C$$$A