Integral de (x^5)*(5-x^2)^1/2 dx

TrigSubstitutionRule(theta=_theta, func=sqrt(5)*sin(_theta), rewritten=125*sqrt(5)*sin(_theta)**5*cos(_theta)**2, substep=ConstantTimesRule(constant=125*sqrt(5), other=sin(_theta)**5*cos(_theta)**2, substep=RewriteRule(rewritten=(1 - cos(_theta)**2)**2*sin(_theta)*cos(_theta)**2, substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=cos(_theta), constant=1, substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=_u**6, substep=PowerRule(base=_u, exp=6, context=_u**6, symbol=_u), context=-_u**6, symbol=_u), ConstantTimesRule(constant=2, other=_u**4, substep=PowerRule(base=_u, exp=4, context=_u**4, symbol=_u), context=2*_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**6 + 2*_u**4 - _u**2, symbol=_u), context=(1 - cos(_theta)**2)**2*sin(_theta)*cos(_theta)**2, symbol=_theta), RewriteRule(rewritten=sin(_theta)*cos(_theta)**6 - 2*sin(_theta)*cos(_theta)**4 + sin(_theta)*cos(_theta)**2, substep=AddRule(substeps=[URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**6, substep=PowerRule(base=_u, exp=6, context=_u**6, symbol=_u), context=_u**6, symbol=_u), context=sin(_theta)*cos(_theta)**6, symbol=_theta), ConstantTimesRule(constant=-2, 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=-2*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)**6 - 2*sin(_theta)*cos(_theta)**4 + sin(_theta)*cos(_theta)**2, symbol=_theta), context=(1 - cos(_theta)**2)**2*sin(_theta)*cos(_theta)**2, symbol=_theta), RewriteRule(rewritten=sin(_theta)*cos(_theta)**6 - 2*sin(_theta)*cos(_theta)**4 + sin(_theta)*cos(_theta)**2, substep=AddRule(substeps=[URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**6, substep=PowerRule(base=_u, exp=6, context=_u**6, symbol=_u), context=_u**6, symbol=_u), context=sin(_theta)*cos(_theta)**6, symbol=_theta), ConstantTimesRule(constant=-2, 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=-2*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)**6 - 2*sin(_theta)*cos(_theta)**4 + sin(_theta)*cos(_theta)**2, symbol=_theta), context=(1 - cos(_theta)**2)**2*sin(_theta)*cos(_theta)**2, symbol=_theta)], context=(1 - cos(_theta)**2)**2*sin(_theta)*cos(_theta)**2, symbol=_theta), context=sin(_theta)**5*cos(_theta)**2, symbol=_theta), context=125*sqrt(5)*sin(_theta)**5*cos(_theta)**2, symbol=_theta), restriction=(x < sqrt(5)) & (x > -sqrt(5)), context=x**5*sqrt(5 - x**2), symbol=x)

1. Ahora simplificar:

   $\begin{cases} \frac{\left(5 - x^{2}\right)^{\frac{3}{2}} \left(- 42 x^{2} - 3 \left(5 - x^{2}\right)^{2} + 35\right)}{21} & \text{for}\: x > - \sqrt{5} \wedge x < \sqrt{5} \end{cases}$

2. Añadimos la constante de integración:

   $\begin{cases} \frac{\left(5 - x^{2}\right)^{\frac{3}{2}} \left(- 42 x^{2} - 3 \left(5 - x^{2}\right)^{2} + 35\right)}{21} & \text{for}\: x > - \sqrt{5} \wedge x < \sqrt{5} \end{cases}+ \mathrm{constant}$

Respuesta:

$\begin{cases} \frac{\left(5 - x^{2}\right)^{\frac{3}{2}} \left(- 42 x^{2} - 3 \left(5 - x^{2}\right)^{2} + 35\right)}{21} & \text{for}\: x > - \sqrt{5} \wedge x < \sqrt{5} \end{cases}+ \mathrm{constant}$