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

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

Para a integral $$$\int{x^{2} \ln{\left(x \right)} d x}$$$, use integração por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

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

Então $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{x^{2} d x}=\frac{x^{3}}{3}$$$ (os passos podem ser vistos »).

A integral pode ser reescrita como

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

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{1}{3}$$$ e $$$f{\left(x \right)} = x^{2}$$$:

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

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=2$$$:

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

Portanto,

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

Simplifique:

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

Adicione a constante de integração:

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

$$$\int x^{2} \ln\left(x\right)\, dx = \frac{x^{3} \left(3 \ln\left(x\right) - 1\right)}{9} + C$$$A