Integral of x^2arcsinx dx

1. Use integration by parts:

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

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

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

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

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

      $$\int x^{2}\, dx = \frac{x^{3}}{3}$$

   Now evaluate the sub-integral.

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

   $$\int \frac{x^{3}}{3 \sqrt{1 - x^{2}}}\, dx = \frac{\int \frac{x^{3}}{\sqrt{1 - x^{2}}}\, dx}{3}$$

   TrigSubstitutionRule(theta=_theta, func=sin(_theta), rewritten=sin(_theta)**3, substep=RewriteRule(rewritten=(1 - cos(_theta)**2)*sin(_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=cos(_theta), constant=1, substep=AddRule(substeps=[PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), ConstantRule(constant=-1, context=-1, symbol=_u)], context=_u**2 - 1, symbol=_u), context=(1 - cos(_theta)**2)*sin(_theta), symbol=_theta), RewriteRule(rewritten=-sin(_theta)*cos(_theta)**2 + sin(_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(_theta)*cos(_theta)**2, substep=URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(_theta)*cos(_theta)**2, symbol=_theta), context=-sin(_theta)*cos(_theta)**2, symbol=_theta), TrigRule(func='sin', arg=_theta, context=sin(_theta), symbol=_theta)], context=-sin(_theta)*cos(_theta)**2 + sin(_theta), symbol=_theta), context=(1 - cos(_theta)**2)*sin(_theta), symbol=_theta), RewriteRule(rewritten=-sin(_theta)*cos(_theta)**2 + sin(_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(_theta)*cos(_theta)**2, substep=URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(_theta)*cos(_theta)**2, symbol=_theta), context=-sin(_theta)*cos(_theta)**2, symbol=_theta), TrigRule(func='sin', arg=_theta, context=sin(_theta), symbol=_theta)], context=-sin(_theta)*cos(_theta)**2 + sin(_theta), symbol=_theta), context=(1 - cos(_theta)**2)*sin(_theta), symbol=_theta)], context=(1 - cos(_theta)**2)*sin(_theta), symbol=_theta), context=sin(_theta)**3, symbol=_theta), restriction=(x > -1) & (x < 1), context=x**3/sqrt(1 - x**2), symbol=x)

   So, the result is: $\frac{\begin{cases} \frac{\left(1 - x^{2}\right)^{\frac{3}{2}}}{3} - \sqrt{1 - x^{2}} & \text{for}\: x > -1 \wedge x < 1 \end{cases}}{3}$

3. Now simplify:

   $$\begin{cases} \frac{x^{3} \operatorname{asin}{\left(x \right)}}{3} + \frac{\sqrt{1 - x^{2}} \left(x^{2} + 2\right)}{9} & \text{for}\: x > -1 \wedge x < 1 \end{cases}$$

4. Add the constant of integration:

   $$\begin{cases} \frac{x^{3} \operatorname{asin}{\left(x \right)}}{3} + \frac{\sqrt{1 - x^{2}} \left(x^{2} + 2\right)}{9} & \text{for}\: x > -1 \wedge x < 1 \end{cases}+ \mathrm{constant}$$

The answer is:

$$\begin{cases} \frac{x^{3} \operatorname{asin}{\left(x \right)}}{3} + \frac{\sqrt{1 - x^{2}} \left(x^{2} + 2\right)}{9} & \text{for}\: x > -1 \wedge x < 1 \end{cases}+ \mathrm{constant}$$