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Resolución de integrales/ 3-x^2/ -x^4/(3*sqrd(3-x^2))

Integral de -x^4/(3*sqrd(3-x^2)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  4                 
  /                 
 |                  
 |         4        
 |       -x         
 |  ------------- dx
 |       ________   
 |      /      2    
 |  3*\/  3 - x     
 |                  
/                   
0
$$\int\limits_{0}^{4} \frac{\left(-1\right) x^{4}}{3 \sqrt{3 - x^{2}}}\, dx$$ 
Integral((-x^4)/((3*sqrt(3 - x^2))), (x, 0, 4))
Solución detallada

    TrigSubstitutionRule(theta=_theta, func=sqrt(3)*sin(_theta), rewritten=-3*sin(_theta)**4, substep=ConstantTimesRule(constant=-3, 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=-3*sin(_theta)**4, symbol=_theta), restriction=(x < sqrt(3)) & (x > -sqrt(3)), context=(-x**4)/((3*sqrt(3 - x**2))), symbol=x)

  1. Ahora simplificar:

  2. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
  /                                                                                                                      
 |                        //        /    ___\                                                                           \
 |        4               ||        |x*\/ 3 |        ________        ________                                           |
 |      -x                ||  9*asin|-------|       /      2        /      2  /       2\                                |
 | ------------- dx = C + |<        \   3   /   x*\/  3 - x     x*\/  3 - x  *\3 - 2*x /         /       ___        ___\|
 |      ________          ||- --------------- + ------------- - ------------------------  for And\x > -\/ 3 , x < \/ 3 /|
 |     /      2           ||         8                2                    24                                           |
 | 3*\/  3 - x            \\                                                                                            /
 |                                                                                                                       
/
$$\int \frac{\left(-1\right) x^{4}}{3 \sqrt{3 - x^{2}}}\, dx = C + \begin{cases} - \frac{x \left(3 - 2 x^{2}\right) \sqrt{3 - x^{2}}}{24} + \frac{x \sqrt{3 - x^{2}}}{2} - \frac{9 \operatorname{asin}{\left(\frac{\sqrt{3} x}{3} \right)}}{8} & \text{for}\: x > - \sqrt{3} \wedge x < \sqrt{3} \end{cases}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
        /    ___\              
        |4*\/ 3 |              
  9*asin|-------|          ____
        \   3   /   41*I*\/ 13 
- --------------- + -----------
         8               6
$$- \frac{9 \operatorname{asin}{\left(\frac{4 \sqrt{3}}{3} \right)}}{8} + \frac{41 \sqrt{13} i}{6}$$ 
=
= 
        /    ___\              
        |4*\/ 3 |              
  9*asin|-------|          ____
        \   3   /   41*I*\/ 13 
- --------------- + -----------
         8               6
$$- \frac{9 \operatorname{asin}{\left(\frac{4 \sqrt{3}}{3} \right)}}{8} + \frac{41 \sqrt{13} i}{6}$$ 
-9*asin(4*sqrt(3)/3)/8 + 41*i*sqrt(13)/6
Respuesta numérica [src] 
(-1.53795339984534 + 26.1034940125881j)
(-1.53795339984534 + 26.1034940125881j)

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

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