$$$\frac{\ln\left(x\right)}{x^{24}}$$$ 的積分

求$$$\int \frac{\ln\left(x\right)}{x^{24}}\, dx$$$。

對於積分 $$$\int{\frac{\ln{\left(x \right)}}{x^{24}} d x}$$$，使用分部積分法 $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

令 $$$\operatorname{u}=\ln{\left(x \right)}$$$ 與 $$$\operatorname{dv}=\frac{dx}{x^{24}}$$$。

則 $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$（步驟見 »），且 $$$\operatorname{v}=\int{\frac{1}{x^{24}} d x}=- \frac{1}{23 x^{23}}$$$（步驟見 »）。

該積分可改寫為

$${\color{red}{\int{\frac{\ln{\left(x \right)}}{x^{24}} d x}}}={\color{red}{\left(\ln{\left(x \right)} \cdot \left(- \frac{1}{23 x^{23}}\right)-\int{\left(- \frac{1}{23 x^{23}}\right) \cdot \frac{1}{x} d x}\right)}}={\color{red}{\left(- \int{\left(- \frac{1}{23 x^{24}}\right)d x} - \frac{\ln{\left(x \right)}}{23 x^{23}}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=- \frac{1}{23}$$$ 與 $$$f{\left(x \right)} = \frac{1}{x^{24}}$$$：

$$- {\color{red}{\int{\left(- \frac{1}{23 x^{24}}\right)d x}}} - \frac{\ln{\left(x \right)}}{23 x^{23}} = - {\color{red}{\left(- \frac{\int{\frac{1}{x^{24}} d x}}{23}\right)}} - \frac{\ln{\left(x \right)}}{23 x^{23}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=-24$$$：

$$\frac{{\color{red}{\int{\frac{1}{x^{24}} d x}}}}{23} - \frac{\ln{\left(x \right)}}{23 x^{23}}=\frac{{\color{red}{\int{x^{-24} d x}}}}{23} - \frac{\ln{\left(x \right)}}{23 x^{23}}=\frac{{\color{red}{\frac{x^{-24 + 1}}{-24 + 1}}}}{23} - \frac{\ln{\left(x \right)}}{23 x^{23}}=\frac{{\color{red}{\left(- \frac{x^{-23}}{23}\right)}}}{23} - \frac{\ln{\left(x \right)}}{23 x^{23}}=\frac{{\color{red}{\left(- \frac{1}{23 x^{23}}\right)}}}{23} - \frac{\ln{\left(x \right)}}{23 x^{23}}$$

因此，

$$\int{\frac{\ln{\left(x \right)}}{x^{24}} d x} = - \frac{\ln{\left(x \right)}}{23 x^{23}} - \frac{1}{529 x^{23}}$$

化簡：

$$\int{\frac{\ln{\left(x \right)}}{x^{24}} d x} = \frac{- 23 \ln{\left(x \right)} - 1}{529 x^{23}}$$

加上積分常數：

$$\int{\frac{\ln{\left(x \right)}}{x^{24}} d x} = \frac{- 23 \ln{\left(x \right)} - 1}{529 x^{23}}+C$$

$$$\int \frac{\ln\left(x\right)}{x^{24}}\, dx = \frac{- 23 \ln\left(x\right) - 1}{529 x^{23}} + C$$$A