Integral of $$$x^{2} \ln^{2}\left(x\right)$$$

Find $$$\int x^{2} \ln^{2}\left(x\right)\, dx$$$.

For the integral $$$\int{x^{2} \ln{\left(x \right)}^{2} d x}$$$, 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 $$$\operatorname{dv}=x^{2} dx$$$.

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 $$$\operatorname{v}=\int{x^{2} d x}=\frac{x^{3}}{3}$$$ (steps can be seen »).

Thus,

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

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

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

For the integral $$$\int{x^{2} \ln{\left(x \right)} d x}$$$, 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 $$$\operatorname{dv}=x^{2} dx$$$.

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

Therefore,

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{3}$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$$\frac{x^{3} \ln{\left(x \right)}^{2}}{3} - \frac{2 x^{3} \ln{\left(x \right)}}{9} + \frac{2 {\color{red}{\int{\frac{x^{2}}{3} d x}}}}{3} = \frac{x^{3} \ln{\left(x \right)}^{2}}{3} - \frac{2 x^{3} \ln{\left(x \right)}}{9} + \frac{2 {\color{red}{\left(\frac{\int{x^{2} d x}}{3}\right)}}}{3}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=2$$$:

$$\frac{x^{3} \ln{\left(x \right)}^{2}}{3} - \frac{2 x^{3} \ln{\left(x \right)}}{9} + \frac{2 {\color{red}{\int{x^{2} d x}}}}{9}=\frac{x^{3} \ln{\left(x \right)}^{2}}{3} - \frac{2 x^{3} \ln{\left(x \right)}}{9} + \frac{2 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}}{9}=\frac{x^{3} \ln{\left(x \right)}^{2}}{3} - \frac{2 x^{3} \ln{\left(x \right)}}{9} + \frac{2 {\color{red}{\left(\frac{x^{3}}{3}\right)}}}{9}$$

Therefore,

$$\int{x^{2} \ln{\left(x \right)}^{2} d x} = \frac{x^{3} \ln{\left(x \right)}^{2}}{3} - \frac{2 x^{3} \ln{\left(x \right)}}{9} + \frac{2 x^{3}}{27}$$

Simplify:

$$\int{x^{2} \ln{\left(x \right)}^{2} d x} = \frac{x^{3} \left(9 \ln{\left(x \right)}^{2} - 6 \ln{\left(x \right)} + 2\right)}{27}$$

Add the constant of integration:

$$\int{x^{2} \ln{\left(x \right)}^{2} d x} = \frac{x^{3} \left(9 \ln{\left(x \right)}^{2} - 6 \ln{\left(x \right)} + 2\right)}{27}+C$$

$$$\int x^{2} \ln^{2}\left(x\right)\, dx = \frac{x^{3} \left(9 \ln^{2}\left(x\right) - 6 \ln\left(x\right) + 2\right)}{27} + C$$$A