$$$\frac{10}{x^{9} \left(x - 1\right)}$$$ 的积分

求$$$\int \frac{10}{x^{9} \left(x - 1\right)}\, dx$$$。

对 $$$c=10$$$ 和 $$$f{\left(x \right)} = \frac{1}{x^{9} \left(x - 1\right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$${\color{red}{\int{\frac{10}{x^{9} \left(x - 1\right)} d x}}} = {\color{red}{\left(10 \int{\frac{1}{x^{9} \left(x - 1\right)} d x}\right)}}$$

进行部分分式分解（步骤可见»）:

$$10 {\color{red}{\int{\frac{1}{x^{9} \left(x - 1\right)} d x}}} = 10 {\color{red}{\int{\left(\frac{1}{x - 1} - \frac{1}{x} - \frac{1}{x^{2}} - \frac{1}{x^{3}} - \frac{1}{x^{4}} - \frac{1}{x^{5}} - \frac{1}{x^{6}} - \frac{1}{x^{7}} - \frac{1}{x^{8}} - \frac{1}{x^{9}}\right)d x}}}$$

逐项积分：

$$10 {\color{red}{\int{\left(\frac{1}{x - 1} - \frac{1}{x} - \frac{1}{x^{2}} - \frac{1}{x^{3}} - \frac{1}{x^{4}} - \frac{1}{x^{5}} - \frac{1}{x^{6}} - \frac{1}{x^{7}} - \frac{1}{x^{8}} - \frac{1}{x^{9}}\right)d x}}} = 10 {\color{red}{\left(- \int{\frac{1}{x^{9}} d x} - \int{\frac{1}{x^{8}} d x} - \int{\frac{1}{x^{7}} d x} - \int{\frac{1}{x^{6}} d x} - \int{\frac{1}{x^{5}} d x} - \int{\frac{1}{x^{4}} d x} - \int{\frac{1}{x^{3}} d x} - \int{\frac{1}{x^{2}} d x} - \int{\frac{1}{x} d x} + \int{\frac{1}{x - 1} d x}\right)}}$$

设$$$u=x - 1$$$。

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

积分变为

$$- 10 \int{\frac{1}{x^{9}} d x} - 10 \int{\frac{1}{x^{8}} d x} - 10 \int{\frac{1}{x^{7}} d x} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 \int{\frac{1}{x} d x} + 10 {\color{red}{\int{\frac{1}{x - 1} d x}}} = - 10 \int{\frac{1}{x^{9}} d x} - 10 \int{\frac{1}{x^{8}} d x} - 10 \int{\frac{1}{x^{7}} d x} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 \int{\frac{1}{x} d x} + 10 {\color{red}{\int{\frac{1}{u} d u}}}$$

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

$$- 10 \int{\frac{1}{x^{9}} d x} - 10 \int{\frac{1}{x^{8}} d x} - 10 \int{\frac{1}{x^{7}} d x} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 \int{\frac{1}{x} d x} + 10 {\color{red}{\int{\frac{1}{u} d u}}} = - 10 \int{\frac{1}{x^{9}} d x} - 10 \int{\frac{1}{x^{8}} d x} - 10 \int{\frac{1}{x^{7}} d x} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 \int{\frac{1}{x} d x} + 10 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

回忆一下 $$$u=x - 1$$$:

$$10 \ln{\left(\left|{{\color{red}{u}}}\right| \right)} - 10 \int{\frac{1}{x^{9}} d x} - 10 \int{\frac{1}{x^{8}} d x} - 10 \int{\frac{1}{x^{7}} d x} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 \int{\frac{1}{x} d x} = 10 \ln{\left(\left|{{\color{red}{\left(x - 1\right)}}}\right| \right)} - 10 \int{\frac{1}{x^{9}} d x} - 10 \int{\frac{1}{x^{8}} d x} - 10 \int{\frac{1}{x^{7}} d x} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 \int{\frac{1}{x} d x}$$

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

$$10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{9}} d x} - 10 \int{\frac{1}{x^{8}} d x} - 10 \int{\frac{1}{x^{7}} d x} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\int{\frac{1}{x} d x}}} = 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{9}} d x} - 10 \int{\frac{1}{x^{8}} d x} - 10 \int{\frac{1}{x^{7}} d x} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=-9$$$：

$$- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{8}} d x} - 10 \int{\frac{1}{x^{7}} d x} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\int{\frac{1}{x^{9}} d x}}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{8}} d x} - 10 \int{\frac{1}{x^{7}} d x} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\int{x^{-9} d x}}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{8}} d x} - 10 \int{\frac{1}{x^{7}} d x} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\frac{x^{-9 + 1}}{-9 + 1}}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{8}} d x} - 10 \int{\frac{1}{x^{7}} d x} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\left(- \frac{x^{-8}}{8}\right)}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{8}} d x} - 10 \int{\frac{1}{x^{7}} d x} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\left(- \frac{1}{8 x^{8}}\right)}}$$

应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=-8$$$：

$$- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{7}} d x} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\int{\frac{1}{x^{8}} d x}}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{7}} d x} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\int{x^{-8} d x}}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{7}} d x} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\frac{x^{-8 + 1}}{-8 + 1}}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{7}} d x} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\left(- \frac{x^{-7}}{7}\right)}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{7}} d x} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\left(- \frac{1}{7 x^{7}}\right)}} + \frac{5}{4 x^{8}}$$

应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=-7$$$：

$$- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\int{\frac{1}{x^{7}} d x}}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\int{x^{-7} d x}}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\frac{x^{-7 + 1}}{-7 + 1}}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\left(- \frac{x^{-6}}{6}\right)}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{6}} d x} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\left(- \frac{1}{6 x^{6}}\right)}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}$$

应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=-6$$$：

$$- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\int{\frac{1}{x^{6}} d x}}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\int{x^{-6} d x}}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\frac{x^{-6 + 1}}{-6 + 1}}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\left(- \frac{x^{-5}}{5}\right)}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{5}} d x} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\left(- \frac{1}{5 x^{5}}\right)}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}$$

应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=-5$$$：

$$- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\int{\frac{1}{x^{5}} d x}}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\int{x^{-5} d x}}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\frac{x^{-5 + 1}}{-5 + 1}}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\left(- \frac{x^{-4}}{4}\right)}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{4}} d x} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\left(- \frac{1}{4 x^{4}}\right)}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}$$

应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=-4$$$：

$$- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\int{\frac{1}{x^{4}} d x}}} + \frac{5}{2 x^{4}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\int{x^{-4} d x}}} + \frac{5}{2 x^{4}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\frac{x^{-4 + 1}}{-4 + 1}}} + \frac{5}{2 x^{4}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\left(- \frac{x^{-3}}{3}\right)}} + \frac{5}{2 x^{4}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{3}} d x} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\left(- \frac{1}{3 x^{3}}\right)}} + \frac{5}{2 x^{4}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}$$

应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=-3$$$：

$$- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\int{\frac{1}{x^{3}} d x}}} + \frac{10}{3 x^{3}} + \frac{5}{2 x^{4}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\int{x^{-3} d x}}} + \frac{10}{3 x^{3}} + \frac{5}{2 x^{4}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\frac{x^{-3 + 1}}{-3 + 1}}} + \frac{10}{3 x^{3}} + \frac{5}{2 x^{4}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\left(- \frac{x^{-2}}{2}\right)}} + \frac{10}{3 x^{3}} + \frac{5}{2 x^{4}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 \int{\frac{1}{x^{2}} d x} - 10 {\color{red}{\left(- \frac{1}{2 x^{2}}\right)}} + \frac{10}{3 x^{3}} + \frac{5}{2 x^{4}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}$$

应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=-2$$$：

$$- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 {\color{red}{\int{\frac{1}{x^{2}} d x}}} + \frac{5}{x^{2}} + \frac{10}{3 x^{3}} + \frac{5}{2 x^{4}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 {\color{red}{\int{x^{-2} d x}}} + \frac{5}{x^{2}} + \frac{10}{3 x^{3}} + \frac{5}{2 x^{4}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 {\color{red}{\frac{x^{-2 + 1}}{-2 + 1}}} + \frac{5}{x^{2}} + \frac{10}{3 x^{3}} + \frac{5}{2 x^{4}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 {\color{red}{\left(- x^{-1}\right)}} + \frac{5}{x^{2}} + \frac{10}{3 x^{3}} + \frac{5}{2 x^{4}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}=- 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} - 10 {\color{red}{\left(- \frac{1}{x}\right)}} + \frac{5}{x^{2}} + \frac{10}{3 x^{3}} + \frac{5}{2 x^{4}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}$$

因此，

$$\int{\frac{10}{x^{9} \left(x - 1\right)} d x} = - 10 \ln{\left(\left|{x}\right| \right)} + 10 \ln{\left(\left|{x - 1}\right| \right)} + \frac{10}{x} + \frac{5}{x^{2}} + \frac{10}{3 x^{3}} + \frac{5}{2 x^{4}} + \frac{2}{x^{5}} + \frac{5}{3 x^{6}} + \frac{10}{7 x^{7}} + \frac{5}{4 x^{8}}$$

化简：

$$\int{\frac{10}{x^{9} \left(x - 1\right)} d x} = \frac{840 x^{8} \left(- \ln{\left(\left|{x}\right| \right)} + \ln{\left(\left|{x - 1}\right| \right)}\right) + 840 x^{7} + 420 x^{6} + 280 x^{5} + 210 x^{4} + 168 x^{3} + 140 x^{2} + 120 x + 105}{84 x^{8}}$$

加上积分常数：

$$\int{\frac{10}{x^{9} \left(x - 1\right)} d x} = \frac{840 x^{8} \left(- \ln{\left(\left|{x}\right| \right)} + \ln{\left(\left|{x - 1}\right| \right)}\right) + 840 x^{7} + 420 x^{6} + 280 x^{5} + 210 x^{4} + 168 x^{3} + 140 x^{2} + 120 x + 105}{84 x^{8}}+C$$

$$$\int \frac{10}{x^{9} \left(x - 1\right)}\, dx = \frac{840 x^{8} \left(- \ln\left(\left|{x}\right|\right) + \ln\left(\left|{x - 1}\right|\right)\right) + 840 x^{7} + 420 x^{6} + 280 x^{5} + 210 x^{4} + 168 x^{3} + 140 x^{2} + 120 x + 105}{84 x^{8}} + C$$$A