Integral of x×arcsin(2x) dx

1. Use integration by parts:

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

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

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

   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.

   SqrtQuadraticDenomRule(a=1, b=0, c=-4, coeffs=[1, 0, 0], context=x**2/sqrt(1 - 4*x**2), symbol=x)

2. Add the constant of integration:

   $$\frac{x^{2} \operatorname{asin}{\left(2 x \right)}}{2} + \frac{x \sqrt{1 - 4 x^{2}}}{8} - \frac{\operatorname{asin}{\left(2 x \right)}}{16}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{2} \operatorname{asin}{\left(2 x \right)}}{2} + \frac{x \sqrt{1 - 4 x^{2}}}{8} - \frac{\operatorname{asin}{\left(2 x \right)}}{16}+ \mathrm{constant}$$