$$$\frac{11 x}{x - 44}$$$ 的积分

求$$$\int \frac{11 x}{x - 44}\, dx$$$。

对 $$$c=11$$$ 和 $$$f{\left(x \right)} = \frac{x}{x - 44}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

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

改写并拆分该分式:

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

逐项积分：

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

应用常数法则 $$$\int c\, dx = c x$$$，使用 $$$c=1$$$：

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

对 $$$c=44$$$ 和 $$$f{\left(x \right)} = \frac{1}{x - 44}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

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

设$$$u=x - 44$$$。

则$$$du=\left(x - 44\right)^{\prime }dx = 1 dx$$$ (步骤见»)，并有$$$dx = du$$$。

因此，

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

$$$\frac{1}{u}$$$ 的积分为 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

回忆一下 $$$u=x - 44$$$:

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

因此，

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

加上积分常数：

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

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