$$$- \frac{24 x}{\left(x - 5\right) \left(x - 3\right)}$$$ 的積分

求$$$\int \left(- \frac{24 x}{\left(x - 5\right) \left(x - 3\right)}\right)\, dx$$$。

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=-24$$$ 與 $$$f{\left(x \right)} = \frac{x}{\left(x - 5\right) \left(x - 3\right)}$$$：

$${\color{red}{\int{\left(- \frac{24 x}{\left(x - 5\right) \left(x - 3\right)}\right)d x}}} = {\color{red}{\left(- 24 \int{\frac{x}{\left(x - 5\right) \left(x - 3\right)} d x}\right)}}$$

進行部分分式分解（步驟可見 »）:

$$- 24 {\color{red}{\int{\frac{x}{\left(x - 5\right) \left(x - 3\right)} d x}}} = - 24 {\color{red}{\int{\left(- \frac{3}{2 \left(x - 3\right)} + \frac{5}{2 \left(x - 5\right)}\right)d x}}}$$

逐項積分：

$$- 24 {\color{red}{\int{\left(- \frac{3}{2 \left(x - 3\right)} + \frac{5}{2 \left(x - 5\right)}\right)d x}}} = - 24 {\color{red}{\left(\int{\frac{5}{2 \left(x - 5\right)} d x} - \int{\frac{3}{2 \left(x - 3\right)} d x}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\frac{3}{2}$$$ 與 $$$f{\left(x \right)} = \frac{1}{x - 3}$$$：

$$- 24 \int{\frac{5}{2 \left(x - 5\right)} d x} + 24 {\color{red}{\int{\frac{3}{2 \left(x - 3\right)} d x}}} = - 24 \int{\frac{5}{2 \left(x - 5\right)} d x} + 24 {\color{red}{\left(\frac{3 \int{\frac{1}{x - 3} d x}}{2}\right)}}$$

令 $$$u=x - 3$$$。

則 $$$du=\left(x - 3\right)^{\prime }dx = 1 dx$$$ (步驟見»)，並可得 $$$dx = du$$$。

因此，

$$- 24 \int{\frac{5}{2 \left(x - 5\right)} d x} + 36 {\color{red}{\int{\frac{1}{x - 3} d x}}} = - 24 \int{\frac{5}{2 \left(x - 5\right)} d x} + 36 {\color{red}{\int{\frac{1}{u} d u}}}$$

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

$$- 24 \int{\frac{5}{2 \left(x - 5\right)} d x} + 36 {\color{red}{\int{\frac{1}{u} d u}}} = - 24 \int{\frac{5}{2 \left(x - 5\right)} d x} + 36 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

回顧一下 $$$u=x - 3$$$：

$$36 \ln{\left(\left|{{\color{red}{u}}}\right| \right)} - 24 \int{\frac{5}{2 \left(x - 5\right)} d x} = 36 \ln{\left(\left|{{\color{red}{\left(x - 3\right)}}}\right| \right)} - 24 \int{\frac{5}{2 \left(x - 5\right)} d x}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\frac{5}{2}$$$ 與 $$$f{\left(x \right)} = \frac{1}{x - 5}$$$：

$$36 \ln{\left(\left|{x - 3}\right| \right)} - 24 {\color{red}{\int{\frac{5}{2 \left(x - 5\right)} d x}}} = 36 \ln{\left(\left|{x - 3}\right| \right)} - 24 {\color{red}{\left(\frac{5 \int{\frac{1}{x - 5} d x}}{2}\right)}}$$

令 $$$u=x - 5$$$。

則 $$$du=\left(x - 5\right)^{\prime }dx = 1 dx$$$ (步驟見»)，並可得 $$$dx = du$$$。

該積分可改寫為

$$36 \ln{\left(\left|{x - 3}\right| \right)} - 60 {\color{red}{\int{\frac{1}{x - 5} d x}}} = 36 \ln{\left(\left|{x - 3}\right| \right)} - 60 {\color{red}{\int{\frac{1}{u} d u}}}$$

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

$$36 \ln{\left(\left|{x - 3}\right| \right)} - 60 {\color{red}{\int{\frac{1}{u} d u}}} = 36 \ln{\left(\left|{x - 3}\right| \right)} - 60 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

回顧一下 $$$u=x - 5$$$：

$$36 \ln{\left(\left|{x - 3}\right| \right)} - 60 \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = 36 \ln{\left(\left|{x - 3}\right| \right)} - 60 \ln{\left(\left|{{\color{red}{\left(x - 5\right)}}}\right| \right)}$$

因此，

$$\int{\left(- \frac{24 x}{\left(x - 5\right) \left(x - 3\right)}\right)d x} = - 60 \ln{\left(\left|{x - 5}\right| \right)} + 36 \ln{\left(\left|{x - 3}\right| \right)}$$

加上積分常數：

$$\int{\left(- \frac{24 x}{\left(x - 5\right) \left(x - 3\right)}\right)d x} = - 60 \ln{\left(\left|{x - 5}\right| \right)} + 36 \ln{\left(\left|{x - 3}\right| \right)}+C$$

$$$\int \left(- \frac{24 x}{\left(x - 5\right) \left(x - 3\right)}\right)\, dx = \left(- 60 \ln\left(\left|{x - 5}\right|\right) + 36 \ln\left(\left|{x - 3}\right|\right)\right) + C$$$A