Integral de x^3/(sqrt(16-x^2)) dx

TrigSubstitutionRule(theta=_theta, func=4*sin(_theta), rewritten=64*sin(_theta)**3, substep=ConstantTimesRule(constant=64, other=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), context=64*sin(_theta)**3, symbol=_theta), restriction=(x > -4) & (x < 4), context=x**3/sqrt(16 - x**2), symbol=x)

1. Ahora simplificar:

   $\begin{cases} - \frac{\sqrt{16 - x^{2}} \left(x^{2} + 32\right)}{3} & \text{for}\: x > -4 \wedge x < 4 \end{cases}$

2. Añadimos la constante de integración:

   $\begin{cases} - \frac{\sqrt{16 - x^{2}} \left(x^{2} + 32\right)}{3} & \text{for}\: x > -4 \wedge x < 4 \end{cases}+ \mathrm{constant}$

Respuesta:

$\begin{cases} - \frac{\sqrt{16 - x^{2}} \left(x^{2} + 32\right)}{3} & \text{for}\: x > -4 \wedge x < 4 \end{cases}+ \mathrm{constant}$