$$$\frac{6}{x^{2} - 22 x}$$$ 的积分

求$$$\int \frac{6}{x^{2} - 22 x}\, dx$$$。

化简被积函数:

$${\color{red}{\int{\frac{6}{x^{2} - 22 x} d x}}} = {\color{red}{\int{\frac{6}{x \left(x - 22\right)} d x}}}$$

对 $$$c=6$$$ 和 $$$f{\left(x \right)} = \frac{1}{x \left(x - 22\right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$${\color{red}{\int{\frac{6}{x \left(x - 22\right)} d x}}} = {\color{red}{\left(6 \int{\frac{1}{x \left(x - 22\right)} d x}\right)}}$$

进行部分分式分解（步骤可见»）:

$$6 {\color{red}{\int{\frac{1}{x \left(x - 22\right)} d x}}} = 6 {\color{red}{\int{\left(\frac{1}{22 \left(x - 22\right)} - \frac{1}{22 x}\right)d x}}}$$

逐项积分：

$$6 {\color{red}{\int{\left(\frac{1}{22 \left(x - 22\right)} - \frac{1}{22 x}\right)d x}}} = 6 {\color{red}{\left(- \int{\frac{1}{22 x} d x} + \int{\frac{1}{22 \left(x - 22\right)} d x}\right)}}$$

对 $$$c=\frac{1}{22}$$$ 和 $$$f{\left(x \right)} = \frac{1}{x}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$6 \int{\frac{1}{22 \left(x - 22\right)} d x} - 6 {\color{red}{\int{\frac{1}{22 x} d x}}} = 6 \int{\frac{1}{22 \left(x - 22\right)} d x} - 6 {\color{red}{\left(\frac{\int{\frac{1}{x} d x}}{22}\right)}}$$

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

$$6 \int{\frac{1}{22 \left(x - 22\right)} d x} - \frac{3 {\color{red}{\int{\frac{1}{x} d x}}}}{11} = 6 \int{\frac{1}{22 \left(x - 22\right)} d x} - \frac{3 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}}{11}$$

对 $$$c=\frac{1}{22}$$$ 和 $$$f{\left(x \right)} = \frac{1}{x - 22}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$- \frac{3 \ln{\left(\left|{x}\right| \right)}}{11} + 6 {\color{red}{\int{\frac{1}{22 \left(x - 22\right)} d x}}} = - \frac{3 \ln{\left(\left|{x}\right| \right)}}{11} + 6 {\color{red}{\left(\frac{\int{\frac{1}{x - 22} d x}}{22}\right)}}$$

设$$$u=x - 22$$$。

则$$$du=\left(x - 22\right)^{\prime }dx = 1 dx$$$ (步骤见»)，并有$$$dx = du$$$。

所以，

$$- \frac{3 \ln{\left(\left|{x}\right| \right)}}{11} + \frac{3 {\color{red}{\int{\frac{1}{x - 22} d x}}}}{11} = - \frac{3 \ln{\left(\left|{x}\right| \right)}}{11} + \frac{3 {\color{red}{\int{\frac{1}{u} d u}}}}{11}$$

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

$$- \frac{3 \ln{\left(\left|{x}\right| \right)}}{11} + \frac{3 {\color{red}{\int{\frac{1}{u} d u}}}}{11} = - \frac{3 \ln{\left(\left|{x}\right| \right)}}{11} + \frac{3 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{11}$$

回忆一下 $$$u=x - 22$$$:

$$- \frac{3 \ln{\left(\left|{x}\right| \right)}}{11} + \frac{3 \ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{11} = - \frac{3 \ln{\left(\left|{x}\right| \right)}}{11} + \frac{3 \ln{\left(\left|{{\color{red}{\left(x - 22\right)}}}\right| \right)}}{11}$$

因此，

$$\int{\frac{6}{x^{2} - 22 x} d x} = - \frac{3 \ln{\left(\left|{x}\right| \right)}}{11} + \frac{3 \ln{\left(\left|{x - 22}\right| \right)}}{11}$$

化简：

$$\int{\frac{6}{x^{2} - 22 x} d x} = \frac{3 \left(- \ln{\left(\left|{x}\right| \right)} + \ln{\left(\left|{x - 22}\right| \right)}\right)}{11}$$

加上积分常数：

$$\int{\frac{6}{x^{2} - 22 x} d x} = \frac{3 \left(- \ln{\left(\left|{x}\right| \right)} + \ln{\left(\left|{x - 22}\right| \right)}\right)}{11}+C$$

$$$\int \frac{6}{x^{2} - 22 x}\, dx = \frac{3 \left(- \ln\left(\left|{x}\right|\right) + \ln\left(\left|{x - 22}\right|\right)\right)}{11} + C$$$A