$$$\frac{11 x}{x - 44}$$$ 的積分

求$$$\int \frac{11 x}{x - 44}\, dx$$$。

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=11$$$ 與 $$$f{\left(x \right)} = \frac{x}{x - 44}$$$：

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

重寫並拆分分式:

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

逐項積分：

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

配合 $$$c=1$$$，應用常數法則 $$$\int c\, dx = c x$$$：

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

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=44$$$ 與 $$$f{\left(x \right)} = \frac{1}{x - 44}$$$：

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

令 $$$u=x - 44$$$。

則 $$$du=\left(x - 44\right)^{\prime }dx = 1 dx$$$ (步驟見»)，並可得 $$$dx = du$$$。

該積分變為

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

$$$\frac{1}{u}$$$ 的積分是 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$：

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

回顧一下 $$$u=x - 44$$$：

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

因此，

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

加上積分常數：

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

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