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

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

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

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

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

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

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

Thus,

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

Simplify the integrand:

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

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

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

Rewrite and split the fraction:

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

Integrate term by term:

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

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

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

The integral of $$$\frac{1}{x^{2} + 1}$$$ is $$$\int{\frac{1}{x^{2} + 1} d x} = \operatorname{atan}{\left(x \right)}$$$:

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

Therefore,

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

Add the constant of integration:

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

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