$$$\ln^{3}\left(x^{6}\right)$$$ 的積分

求$$$\int \ln^{3}\left(x^{6}\right)\, dx$$$。

已將輸入重寫為：$$$\int{\ln{\left(x^{6} \right)}^{3} d x}=\int{216 \ln{\left(x \right)}^{3} d x}$$$。

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=216$$$ 與 $$$f{\left(x \right)} = \ln{\left(x \right)}^{3}$$$：

$${\color{red}{\int{216 \ln{\left(x \right)}^{3} d x}}} = {\color{red}{\left(216 \int{\ln{\left(x \right)}^{3} d x}\right)}}$$

對於積分 $$$\int{\ln{\left(x \right)}^{3} d x}$$$，使用分部積分法 $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

令 $$$\operatorname{u}=\ln{\left(x \right)}^{3}$$$ 與 $$$\operatorname{dv}=dx$$$。

則 $$$\operatorname{du}=\left(\ln{\left(x \right)}^{3}\right)^{\prime }dx=\frac{3 \ln{\left(x \right)}^{2}}{x} dx$$$（步驟見 »），且 $$$\operatorname{v}=\int{1 d x}=x$$$（步驟見 »）。

該積分變為

$$216 {\color{red}{\int{\ln{\left(x \right)}^{3} d x}}}=216 {\color{red}{\left(\ln{\left(x \right)}^{3} \cdot x-\int{x \cdot \frac{3 \ln{\left(x \right)}^{2}}{x} d x}\right)}}=216 {\color{red}{\left(x \ln{\left(x \right)}^{3} - \int{3 \ln{\left(x \right)}^{2} d x}\right)}}$$

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

$$216 x \ln{\left(x \right)}^{3} - 216 {\color{red}{\int{3 \ln{\left(x \right)}^{2} d x}}} = 216 x \ln{\left(x \right)}^{3} - 216 {\color{red}{\left(3 \int{\ln{\left(x \right)}^{2} d x}\right)}}$$

對於積分 $$$\int{\ln{\left(x \right)}^{2} d x}$$$，使用分部積分法 $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

令 $$$\operatorname{u}=\ln{\left(x \right)}^{2}$$$ 與 $$$\operatorname{dv}=dx$$$。

則 $$$\operatorname{du}=\left(\ln{\left(x \right)}^{2}\right)^{\prime }dx=\frac{2 \ln{\left(x \right)}}{x} dx$$$（步驟見 »），且 $$$\operatorname{v}=\int{1 d x}=x$$$（步驟見 »）。

所以，

$$216 x \ln{\left(x \right)}^{3} - 648 {\color{red}{\int{\ln{\left(x \right)}^{2} d x}}}=216 x \ln{\left(x \right)}^{3} - 648 {\color{red}{\left(\ln{\left(x \right)}^{2} \cdot x-\int{x \cdot \frac{2 \ln{\left(x \right)}}{x} d x}\right)}}=216 x \ln{\left(x \right)}^{3} - 648 {\color{red}{\left(x \ln{\left(x \right)}^{2} - \int{2 \ln{\left(x \right)} d x}\right)}}$$

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

$$216 x \ln{\left(x \right)}^{3} - 648 x \ln{\left(x \right)}^{2} + 648 {\color{red}{\int{2 \ln{\left(x \right)} d x}}} = 216 x \ln{\left(x \right)}^{3} - 648 x \ln{\left(x \right)}^{2} + 648 {\color{red}{\left(2 \int{\ln{\left(x \right)} d x}\right)}}$$

對於積分 $$$\int{\ln{\left(x \right)} d x}$$$，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 $$$\operatorname{dv}=dx$$$。

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

因此，

$$216 x \ln{\left(x \right)}^{3} - 648 x \ln{\left(x \right)}^{2} + 1296 {\color{red}{\int{\ln{\left(x \right)} d x}}}=216 x \ln{\left(x \right)}^{3} - 648 x \ln{\left(x \right)}^{2} + 1296 {\color{red}{\left(\ln{\left(x \right)} \cdot x-\int{x \cdot \frac{1}{x} d x}\right)}}=216 x \ln{\left(x \right)}^{3} - 648 x \ln{\left(x \right)}^{2} + 1296 {\color{red}{\left(x \ln{\left(x \right)} - \int{1 d x}\right)}}$$

配合 $$$c=1$$$，應用常數法則 $$$\int c\, dx = c x$$$：

$$216 x \ln{\left(x \right)}^{3} - 648 x \ln{\left(x \right)}^{2} + 1296 x \ln{\left(x \right)} - 1296 {\color{red}{\int{1 d x}}} = 216 x \ln{\left(x \right)}^{3} - 648 x \ln{\left(x \right)}^{2} + 1296 x \ln{\left(x \right)} - 1296 {\color{red}{x}}$$

因此，

$$\int{216 \ln{\left(x \right)}^{3} d x} = 216 x \ln{\left(x \right)}^{3} - 648 x \ln{\left(x \right)}^{2} + 1296 x \ln{\left(x \right)} - 1296 x$$

化簡：

$$\int{216 \ln{\left(x \right)}^{3} d x} = 216 x \left(\ln{\left(x \right)}^{3} - 3 \ln{\left(x \right)}^{2} + 6 \ln{\left(x \right)} - 6\right)$$

加上積分常數：

$$\int{216 \ln{\left(x \right)}^{3} d x} = 216 x \left(\ln{\left(x \right)}^{3} - 3 \ln{\left(x \right)}^{2} + 6 \ln{\left(x \right)} - 6\right)+C$$

$$$\int \ln^{3}\left(x^{6}\right)\, dx = 216 x \left(\ln^{3}\left(x\right) - 3 \ln^{2}\left(x\right) + 6 \ln\left(x\right) - 6\right) + C$$$A