Integral de X^3sqrt(25-x^2) dx

TrigSubstitutionRule(theta=_theta, func=5*sin(_theta), rewritten=3125*sin(_theta)**3*cos(_theta)**2, substep=ConstantTimesRule(constant=3125, other=sin(_theta)**3*cos(_theta)**2, substep=RewriteRule(rewritten=(1 - cos(_theta)**2)*sin(_theta)*cos(_theta)**2, substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=cos(_theta), constant=1, substep=AddRule(substeps=[PowerRule(base=_u, exp=4, context=_u**4, symbol=_u), ConstantTimesRule(constant=-1, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=-_u**2, symbol=_u)], context=_u**4 - _u**2, symbol=_u), context=(1 - cos(_theta)**2)*sin(_theta)*cos(_theta)**2, symbol=_theta), RewriteRule(rewritten=-sin(_theta)*cos(_theta)**4 + sin(_theta)*cos(_theta)**2, substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(_theta)*cos(_theta)**4, substep=URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**4, substep=PowerRule(base=_u, exp=4, context=_u**4, symbol=_u), context=_u**4, symbol=_u), context=sin(_theta)*cos(_theta)**4, symbol=_theta), context=-sin(_theta)*cos(_theta)**4, symbol=_theta), 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)**4 + sin(_theta)*cos(_theta)**2, symbol=_theta), context=(1 - cos(_theta)**2)*sin(_theta)*cos(_theta)**2, symbol=_theta), RewriteRule(rewritten=-sin(_theta)*cos(_theta)**4 + sin(_theta)*cos(_theta)**2, substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(_theta)*cos(_theta)**4, substep=URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**4, substep=PowerRule(base=_u, exp=4, context=_u**4, symbol=_u), context=_u**4, symbol=_u), context=sin(_theta)*cos(_theta)**4, symbol=_theta), context=-sin(_theta)*cos(_theta)**4, symbol=_theta), 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)**4 + sin(_theta)*cos(_theta)**2, symbol=_theta), context=(1 - cos(_theta)**2)*sin(_theta)*cos(_theta)**2, symbol=_theta)], context=(1 - cos(_theta)**2)*sin(_theta)*cos(_theta)**2, symbol=_theta), context=sin(_theta)**3*cos(_theta)**2, symbol=_theta), context=3125*sin(_theta)**3*cos(_theta)**2, symbol=_theta), restriction=(x > -5) & (x < 5), context=x**3*sqrt(25 - x**2), symbol=x)

1. Ahora simplificar:

   $\begin{cases} - \frac{\left(25 - x^{2}\right)^{\frac{3}{2}} \left(3 x^{2} + 50\right)}{15} & \text{for}\: x > -5 \wedge x < 5 \end{cases}$

2. Añadimos la constante de integración:

   $\begin{cases} - \frac{\left(25 - x^{2}\right)^{\frac{3}{2}} \left(3 x^{2} + 50\right)}{15} & \text{for}\: x > -5 \wedge x < 5 \end{cases}+ \mathrm{constant}$

Respuesta:

$\begin{cases} - \frac{\left(25 - x^{2}\right)^{\frac{3}{2}} \left(3 x^{2} + 50\right)}{15} & \text{for}\: x > -5 \wedge x < 5 \end{cases}+ \mathrm{constant}$