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Resolución de integrales/ (9-x²)/ x³sqrt(9-x²)

Integral de x³sqrt(9-x²) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                  
  /                  
 |                   
 |        ________   
 |   3   /      2    
 |  x *\/  9 - x   dx
 |                   
/                    
0
$$\int\limits_{0}^{1} x^{3} \sqrt{9 - x^{2}}\, dx$$ 
Integral(x^3*sqrt(9 - x^2), (x, 0, 1))
Solución detallada

    TrigSubstitutionRule(theta=_theta, func=3*sin(_theta), rewritten=243*sin(_theta)**3*cos(_theta)**2, substep=ConstantTimesRule(constant=243, 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=243*sin(_theta)**3*cos(_theta)**2, symbol=_theta), restriction=(x > -3) & (x < 3), context=x**3*sqrt(9 - x**2), symbol=x)

  1. Ahora simplificar:

  2. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
  /                                                                                
 |                                                                                 
 |       ________          //                          5/2                        \
 |  3   /      2           ||            3/2   /     2\                           |
 | x *\/  9 - x   dx = C + |<    /     2\      \9 - x /                           |
 |                         ||- 3*\9 - x /    + -----------  for And(x > -3, x < 3)|
/                          \\                       5                             /
$$\int x^{3} \sqrt{9 - x^{2}}\, dx = C + \begin{cases} \frac{\left(9 - x^{2}\right)^{\frac{5}{2}}}{5} - 3 \left(9 - x^{2}\right)^{\frac{3}{2}} & \text{for}\: x > -3 \wedge x < 3 \end{cases}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
            ___
162   112*\/ 2 
--- - ---------
 5        5
$$\frac{162}{5} - \frac{112 \sqrt{2}}{5}$$ 
=
= 
            ___
162   112*\/ 2 
--- - ---------
 5        5
$$\frac{162}{5} - \frac{112 \sqrt{2}}{5}$$ 
162/5 - 112*sqrt(2)/5
Respuesta numérica [src] 
0.721616202842671
0.721616202842671

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

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