Integral dari $$$\frac{11 x}{x - 44}$$$

Temukan $$$\int \frac{11 x}{x - 44}\, dx$$$.

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=11$$$ dan $$$f{\left(x \right)} = \frac{x}{x - 44}$$$:

$${\color{red}{\int{\frac{11 x}{x - 44} d x}}} = {\color{red}{\left(11 \int{\frac{x}{x - 44} d x}\right)}}$$

Tulis ulang dan pisahkan pecahannya:

$$11 {\color{red}{\int{\frac{x}{x - 44} d x}}} = 11 {\color{red}{\int{\left(1 + \frac{44}{x - 44}\right)d x}}}$$

Integralkan suku demi suku:

$$11 {\color{red}{\int{\left(1 + \frac{44}{x - 44}\right)d x}}} = 11 {\color{red}{\left(\int{1 d x} + \int{\frac{44}{x - 44} d x}\right)}}$$

Terapkan aturan konstanta $$$\int c\, dx = c x$$$ dengan $$$c=1$$$:

$$11 \int{\frac{44}{x - 44} d x} + 11 {\color{red}{\int{1 d x}}} = 11 \int{\frac{44}{x - 44} d x} + 11 {\color{red}{x}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=44$$$ dan $$$f{\left(x \right)} = \frac{1}{x - 44}$$$:

$$11 x + 11 {\color{red}{\int{\frac{44}{x - 44} d x}}} = 11 x + 11 {\color{red}{\left(44 \int{\frac{1}{x - 44} d x}\right)}}$$

Misalkan $$$u=x - 44$$$.

Kemudian $$$du=\left(x - 44\right)^{\prime }dx = 1 dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$dx = du$$$.

Dengan demikian,

$$11 x + 484 {\color{red}{\int{\frac{1}{x - 44} d x}}} = 11 x + 484 {\color{red}{\int{\frac{1}{u} d u}}}$$

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

$$11 x + 484 {\color{red}{\int{\frac{1}{u} d u}}} = 11 x + 484 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

Ingat bahwa $$$u=x - 44$$$:

$$11 x + 484 \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = 11 x + 484 \ln{\left(\left|{{\color{red}{\left(x - 44\right)}}}\right| \right)}$$

Oleh karena itu,

$$\int{\frac{11 x}{x - 44} d x} = 11 x + 484 \ln{\left(\left|{x - 44}\right| \right)}$$

Tambahkan konstanta integrasi:

$$\int{\frac{11 x}{x - 44} d x} = 11 x + 484 \ln{\left(\left|{x - 44}\right| \right)}+C$$

$$$\int \frac{11 x}{x - 44}\, dx = \left(11 x + 484 \ln\left(\left|{x - 44}\right|\right)\right) + C$$$A