Integral de $$$2 x \operatorname{atan}{\left(x \right)}$$$

Encontre $$$\int 2 x \operatorname{atan}{\left(x \right)}\, dx$$$.

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

$${\color{red}{\int{2 x \operatorname{atan}{\left(x \right)} d x}}} = {\color{red}{\left(2 \int{x \operatorname{atan}{\left(x \right)} d x}\right)}}$$

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

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

Então $$$\operatorname{du}=\left(\operatorname{atan}{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x^{2} + 1}$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{x d x}=\frac{x^{2}}{2}$$$ (os passos podem ser vistos »).

Logo,

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

Simplifique o integrando:

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

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

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

Reescreva e separe a fração:

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

Integre termo a termo:

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

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=1$$$:

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

A integral de $$$\frac{1}{x^{2} + 1}$$$ é $$$\int{\frac{1}{x^{2} + 1} d x} = \operatorname{atan}{\left(x \right)}$$$:

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

Portanto,

$$\int{2 x \operatorname{atan}{\left(x \right)} d x} = x^{2} \operatorname{atan}{\left(x \right)} - x + \operatorname{atan}{\left(x \right)}$$

Adicione a constante de integração:

$$\int{2 x \operatorname{atan}{\left(x \right)} d x} = x^{2} \operatorname{atan}{\left(x \right)} - x + \operatorname{atan}{\left(x \right)}+C$$

$$$\int 2 x \operatorname{atan}{\left(x \right)}\, dx = \left(x^{2} \operatorname{atan}{\left(x \right)} - x + \operatorname{atan}{\left(x \right)}\right) + C$$$A