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Resolución de integrales/ 4+x^2/ 1/((4+x^2)^3)

Integral de 1/((4+x^2)^3) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

    TrigSubstitutionRule(theta=_theta, func=2*tan(_theta), rewritten=cos(_theta)**4/32, substep=ConstantTimesRule(constant=1/32, other=cos(_theta)**4, substep=RewriteRule(rewritten=(cos(2*_theta)/2 + 1/2)**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=(cos(2*_theta)/2 + 1/2)**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=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta)], context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta), context=cos(_theta)**4, symbol=_theta), context=cos(_theta)**4/32, symbol=_theta), restriction=True, context=1/((x**2 + 4)**3), symbol=x)

  1. Ahora simplificar:

  2. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
  /                         /x\                              
 |                    3*atan|-|                     /     2\ 
 |     1                    \2/        x          x*\4 - x / 
 | --------- dx = C + --------- + ----------- + -------------
 |         3             256         /     2\               2
 | /     2\                       32*\4 + x /       /     2\ 
 | \4 + x /                                     128*\4 + x / 
 |                                                           
/
$$\int \frac{1}{\left(x^{2} + 4\right)^{3}}\, dx = C + \frac{x \left(4 - x^{2}\right)}{128 \left(x^{2} + 4\right)^{2}} + \frac{x}{32 \left(x^{2} + 4\right)} + \frac{3 \operatorname{atan}{\left(\frac{x}{2} \right)}}{256}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
 23    3*atan(1/2)
---- + -----------
3200       256
$$\frac{3 \operatorname{atan}{\left(\frac{1}{2} \right)}}{256} + \frac{23}{3200}$$ 
=
= 
 23    3*atan(1/2)
---- + -----------
3200       256
$$\frac{3 \operatorname{atan}{\left(\frac{1}{2} \right)}}{256} + \frac{23}{3200}$$ 
23/3200 + 3*atan(1/2)/256
Respuesta numérica [src] 
0.0126208704179782
0.0126208704179782

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

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