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Resolución de integrales/ 16-x^2/ x^4/(16-x^2)(sqrt(16-x^2))

Integral de x^4/(16-x^2)(sqrt(16-x^2)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
      ___                       
 -2*\/ 2                        
     /                          
    |                           
    |        4      _________   
    |       x      /       2    
    |    -------*\/  16 - x   dx
    |          2                
    |    16 - x                 
    |                           
   /                            
   0
$$\int\limits_{0}^{- 2 \sqrt{2}} \frac{x^{4}}{16 - x^{2}} \sqrt{16 - x^{2}}\, dx$$ 
Integral((x^4/(16 - x^2))*sqrt(16 - x^2), (x, 0, -2*sqrt(2)))
Solución detallada

    TrigSubstitutionRule(theta=_theta, func=4*sin(_theta), rewritten=256*sin(_theta)**4, substep=ConstantTimesRule(constant=256, other=sin(_theta)**4, substep=RewriteRule(rewritten=(1/2 - cos(2*_theta)/2)**2, substep=AlternativeRule(alternatives=[RewriteRule(rewritten=cos(2*_theta)**2/4 - cos(2*_theta)/2 + 1/4, substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=cos(2*_theta)**2, substep=RewriteRule(rewritten=cos(4*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(4*_theta), substep=URule(u_var=_u, u_func=4*_theta, constant=1/4, substep=ConstantTimesRule(constant=1/4, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(4*_theta), symbol=_theta), context=cos(4*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(4*_theta)/2 + 1/2, symbol=_theta), context=cos(2*_theta)**2, symbol=_theta), context=cos(2*_theta)**2/4, symbol=_theta), ConstantTimesRule(constant=-1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=-cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/4, context=1/4, symbol=_theta)], context=cos(2*_theta)**2/4 - cos(2*_theta)/2 + 1/4, symbol=_theta), context=(1/2 - cos(2*_theta)/2)**2, symbol=_theta), RewriteRule(rewritten=cos(2*_theta)**2/4 - cos(2*_theta)/2 + 1/4, substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=cos(2*_theta)**2, substep=RewriteRule(rewritten=cos(4*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(4*_theta), substep=URule(u_var=_u, u_func=4*_theta, constant=1/4, substep=ConstantTimesRule(constant=1/4, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(4*_theta), symbol=_theta), context=cos(4*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(4*_theta)/2 + 1/2, symbol=_theta), context=cos(2*_theta)**2, symbol=_theta), context=cos(2*_theta)**2/4, symbol=_theta), ConstantTimesRule(constant=-1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=-cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/4, context=1/4, symbol=_theta)], context=cos(2*_theta)**2/4 - cos(2*_theta)/2 + 1/4, symbol=_theta), context=(1/2 - cos(2*_theta)/2)**2, symbol=_theta)], context=(1/2 - cos(2*_theta)/2)**2, symbol=_theta), context=sin(_theta)**4, symbol=_theta), context=256*sin(_theta)**4, symbol=_theta), restriction=(x > -4) & (x < 4), context=(x**4/(16 - x**2))*sqrt(16 - x**2), symbol=x)

  1. Ahora simplificar:

  2. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
  /                                                                                                                
 |                                                                                                                 
 |     4      _________          //                                     _________                                 \
 |    x      /       2           ||                    _________       /       2  /     2\                        |
 | -------*\/  16 - x   dx = C + |<       /x\         /       2    x*\/  16 - x  *\8 - x /                        |
 |       2                       ||96*asin|-| - 8*x*\/  16 - x   + -----------------------  for And(x > -4, x < 4)|
 | 16 - x                        \\       \4/                                 4                                   /
 |                                                                                                                 
/
$$\int \frac{x^{4}}{16 - x^{2}} \sqrt{16 - x^{2}}\, dx = C + \begin{cases} \frac{x \left(8 - x^{2}\right) \sqrt{16 - x^{2}}}{4} - 8 x \sqrt{16 - x^{2}} + 96 \operatorname{asin}{\left(\frac{x}{4} \right)} & \text{for}\: x > -4 \wedge x < 4 \end{cases}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
64 - 24*pi
$$64 - 24 \pi$$ 
=
= 
64 - 24*pi
$$64 - 24 \pi$$ 
64 - 24*pi
Respuesta numérica [src] 
-11.398223686155
-11.398223686155

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

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