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Resolución de integrales/ x^2-4/ x^5sqrt(x^2-4)

Integral de x^5sqrt(x^2-4) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                  
  /                  
 |                   
 |        ________   
 |   5   /  2        
 |  x *\/  x  - 4  dx
 |                   
/                    
0
$$\int\limits_{0}^{1} x^{5} \sqrt{x^{2} - 4}\, dx$$ 
Integral(x^5*sqrt(x^2 - 4), (x, 0, 1))
Solución detallada

    TrigSubstitutionRule(theta=_theta, func=2*sec(_theta), rewritten=128*tan(_theta)**2*sec(_theta)**6, substep=ConstantTimesRule(constant=128, other=tan(_theta)**2*sec(_theta)**6, substep=RewriteRule(rewritten=(tan(_theta)**2 + 1)**2*tan(_theta)**2*sec(_theta)**2, substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=tan(_theta), constant=1, substep=AddRule(substeps=[PowerRule(base=_u, exp=6, 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), PowerRule(base=_u, exp=2, context=_u**2, symbol=_u)], context=_u**6 + 2*_u**4 + _u**2, symbol=_u), context=(tan(_theta)**2 + 1)**2*tan(_theta)**2*sec(_theta)**2, symbol=_theta), RewriteRule(rewritten=tan(_theta)**6*sec(_theta)**2 + 2*tan(_theta)**4*sec(_theta)**2 + tan(_theta)**2*sec(_theta)**2, substep=AddRule(substeps=[URule(u_var=_u, u_func=tan(_theta), constant=1, substep=PowerRule(base=_u, exp=6, context=_u**6, symbol=_u), context=tan(_theta)**6*sec(_theta)**2, symbol=_theta), ConstantTimesRule(constant=2, other=tan(_theta)**4*sec(_theta)**2, substep=URule(u_var=_u, u_func=tan(_theta), constant=1, substep=PowerRule(base=_u, exp=4, context=_u**4, symbol=_u), context=tan(_theta)**4*sec(_theta)**2, symbol=_theta), context=2*tan(_theta)**4*sec(_theta)**2, symbol=_theta), URule(u_var=_u, u_func=tan(_theta), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=tan(_theta)**2*sec(_theta)**2, symbol=_theta)], context=tan(_theta)**6*sec(_theta)**2 + 2*tan(_theta)**4*sec(_theta)**2 + tan(_theta)**2*sec(_theta)**2, symbol=_theta), context=(tan(_theta)**2 + 1)**2*tan(_theta)**2*sec(_theta)**2, symbol=_theta), RewriteRule(rewritten=tan(_theta)**6*sec(_theta)**2 + 2*tan(_theta)**4*sec(_theta)**2 + tan(_theta)**2*sec(_theta)**2, substep=AddRule(substeps=[URule(u_var=_u, u_func=tan(_theta), constant=1, substep=PowerRule(base=_u, exp=6, context=_u**6, symbol=_u), context=tan(_theta)**6*sec(_theta)**2, symbol=_theta), ConstantTimesRule(constant=2, other=tan(_theta)**4*sec(_theta)**2, substep=URule(u_var=_u, u_func=tan(_theta), constant=1, substep=PowerRule(base=_u, exp=4, context=_u**4, symbol=_u), context=tan(_theta)**4*sec(_theta)**2, symbol=_theta), context=2*tan(_theta)**4*sec(_theta)**2, symbol=_theta), URule(u_var=_u, u_func=tan(_theta), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=tan(_theta)**2*sec(_theta)**2, symbol=_theta)], context=tan(_theta)**6*sec(_theta)**2 + 2*tan(_theta)**4*sec(_theta)**2 + tan(_theta)**2*sec(_theta)**2, symbol=_theta), context=(tan(_theta)**2 + 1)**2*tan(_theta)**2*sec(_theta)**2, symbol=_theta)], context=(tan(_theta)**2 + 1)**2*tan(_theta)**2*sec(_theta)**2, symbol=_theta), context=tan(_theta)**2*sec(_theta)**6, symbol=_theta), context=128*tan(_theta)**2*sec(_theta)**6, symbol=_theta), restriction=(x > -2) & (x < 2), context=x**5*sqrt(x**2 - 4), symbol=x)

  1. Ahora simplificar:

  2. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
  /                                                                                                  
 |                                                                                                   
 |       ________          //         7/2              5/2               3/2                        \
 |  5   /  2               ||/      2\        /      2\         /      2\                           |
 | x *\/  x  - 4  dx = C + |<\-4 + x /      8*\-4 + x /      16*\-4 + x /                           |
 |                         ||------------ + -------------- + ---------------  for And(x > -2, x < 2)|
/                          \\     7               5                 3                               /
$$\int x^{5} \sqrt{x^{2} - 4}\, dx = C + \begin{cases} \frac{\left(x^{2} - 4\right)^{\frac{7}{2}}}{7} + \frac{8 \left(x^{2} - 4\right)^{\frac{5}{2}}}{5} + \frac{16 \left(x^{2} - 4\right)^{\frac{3}{2}}}{3} & \text{for}\: x > -2 \wedge x < 2 \end{cases}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                 ___
1024*I   191*I*\/ 3 
------ - -----------
 105          35
$$- \frac{191 \sqrt{3} i}{35} + \frac{1024 i}{105}$$ 
=
= 
                 ___
1024*I   191*I*\/ 3 
------ - -----------
 105          35
$$- \frac{191 \sqrt{3} i}{35} + \frac{1024 i}{105}$$ 
1024*i/105 - 191*i*sqrt(3)/35
Respuesta numérica [src] 
(0.0 + 0.300332259647936j)
(0.0 + 0.300332259647936j)

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

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