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Resolución de integrales/ 5-x^2/ x^3*(25-x^2)^0.5

Integral de x^3*(25-x^2)^0.5 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  5                   
  /                   
 |                    
 |        _________   
 |   3   /       2    
 |  x *\/  25 - x   dx
 |                    
/                     
0
$$\int\limits_{0}^{5} x^{3} \sqrt{25 - x^{2}}\, dx$$ 
Integral(x^3*sqrt(25 - x^2), (x, 0, 5))
Solución detallada

    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:

  2. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
  /                                                                                    
 |                                                                                     
 |       _________          //              3/2            5/2                        \
 |  3   /       2           ||     /      2\      /      2\                           |
 | x *\/  25 - x   dx = C + |<  25*\25 - x /      \25 - x /                           |
 |                          ||- --------------- + ------------  for And(x > -5, x < 5)|
/                           \\         3               5                              /
$$\int x^{3} \sqrt{25 - x^{2}}\, dx = C + \begin{cases} \frac{\left(25 - x^{2}\right)^{\frac{5}{2}}}{5} - \frac{25 \left(25 - x^{2}\right)^{\frac{3}{2}}}{3} & \text{for}\: x > -5 \wedge x < 5 \end{cases}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
1250/3
$$\frac{1250}{3}$$ 
=
= 
1250/3
$$\frac{1250}{3}$$ 
1250/3
Respuesta numérica [src] 
416.666666666667
416.666666666667

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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