Integral of x*arctg(2x) dx

1. Use integration by parts:

   $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

   Let $u{\left(x \right)} = \operatorname{atan}{\left(2 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = x$.

   Then $\operatorname{du}{\left(x \right)} = \frac{2}{4 x^{2} + 1}$.

   To find $v{\left(x \right)}$:

   1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

      $$\int x\, dx = \frac{x^{2}}{2}$$

   Now evaluate the sub-integral.

2. Rewrite the integrand:

   $$\frac{x^{2}}{4 x^{2} + 1} = \frac{1}{4} - \frac{1}{4 \cdot \left(4 x^{2} + 1\right)}$$

3. Integrate term-by-term:

   1. The integral of a constant is the constant times the variable of integration:

      $$\int \frac{1}{4}\, dx = \frac{x}{4}$$

   1. The integral of a constant times a function is the constant times the integral of the function:

      $$\int \left(- \frac{1}{4 \cdot \left(4 x^{2} + 1\right)}\right)\, dx = - \frac{\int \frac{1}{4 x^{2} + 1}\, dx}{4}$$

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

      So, the result is: $- \frac{\operatorname{atan}{\left(2 x \right)}}{8}$

   The result is: $\frac{x}{4} - \frac{\operatorname{atan}{\left(2 x \right)}}{8}$

4. Add the constant of integration:

   $$\frac{x^{2} \operatorname{atan}{\left(2 x \right)}}{2} - \frac{x}{4} + \frac{\operatorname{atan}{\left(2 x \right)}}{8}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{2} \operatorname{atan}{\left(2 x \right)}}{2} - \frac{x}{4} + \frac{\operatorname{atan}{\left(2 x \right)}}{8}+ \mathrm{constant}$$