Integral of $$$\ln^{3}\left(x^{6}\right)$$$

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

The input is rewritten: $$$\int{\ln{\left(x^{6} \right)}^{3} d x}=\int{216 \ln{\left(x \right)}^{3} d x}$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=216$$$ and $$$f{\left(x \right)} = \ln{\left(x \right)}^{3}$$$:

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

For the integral $$$\int{\ln{\left(x \right)}^{3} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=\ln{\left(x \right)}^{3}$$$ and $$$\operatorname{dv}=dx$$$.

Then $$$\operatorname{du}=\left(\ln{\left(x \right)}^{3}\right)^{\prime }dx=\frac{3 \ln{\left(x \right)}^{2}}{x} dx$$$ (steps can be seen ») and $$$\operatorname{v}=\int{1 d x}=x$$$ (steps can be seen »).

So,

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=3$$$ and $$$f{\left(x \right)} = \ln{\left(x \right)}^{2}$$$:

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

For the integral $$$\int{\ln{\left(x \right)}^{2} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=\ln{\left(x \right)}^{2}$$$ and $$$\operatorname{dv}=dx$$$.

Then $$$\operatorname{du}=\left(\ln{\left(x \right)}^{2}\right)^{\prime }dx=\frac{2 \ln{\left(x \right)}}{x} dx$$$ (steps can be seen ») and $$$\operatorname{v}=\int{1 d x}=x$$$ (steps can be seen »).

The integral becomes

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=2$$$ and $$$f{\left(x \right)} = \ln{\left(x \right)}$$$:

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

For the integral $$$\int{\ln{\left(x \right)} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=\ln{\left(x \right)}$$$ and $$$\operatorname{dv}=dx$$$.

Then $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (steps can be seen ») and $$$\operatorname{v}=\int{1 d x}=x$$$ (steps can be seen »).

Therefore,

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

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=1$$$:

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

Therefore,

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

Simplify:

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

Add the constant of integration:

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

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