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

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

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=3$$$ dan $$$f{\left(x \right)} = \frac{1}{3 x - 1}$$$:

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

Misalkan $$$u=3 x - 1$$$.

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

Dengan demikian,

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

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=\frac{1}{3}$$$ dan $$$f{\left(u \right)} = \frac{1}{u}$$$:

$$3 {\color{red}{\int{\frac{1}{3 u} d u}}} = 3 {\color{red}{\left(\frac{\int{\frac{1}{u} d u}}{3}\right)}}$$

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 $$$u=3 x - 1$$$:

$$\ln{\left(\left|{{\color{red}{u}}}\right| \right)} = \ln{\left(\left|{{\color{red}{\left(3 x - 1\right)}}}\right| \right)}$$

Oleh karena itu,

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

Tambahkan konstanta integrasi:

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

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