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Resolución de integrales/ 9+x^2/ 1/((9+x^2)^3)

Integral de 1/((9+x^2)^3) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

    TrigSubstitutionRule(theta=_theta, func=3*tan(_theta), rewritten=cos(_theta)**4/243, substep=ConstantTimesRule(constant=1/243, 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)**4/243, symbol=_theta), restriction=True, context=1/((x**2 + 9)**3), symbol=x)

  1. Ahora simplificar:

  2. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
  /                       /x\                               
 |                    atan|-|                      /     2\ 
 |     1                  \3/        x           x*\9 - x / 
 | --------- dx = C + ------- + ------------ + -------------
 |         3            648         /     2\               2
 | /     2\                     162*\9 + x /       /     2\ 
 | \9 + x /                                    648*\9 + x / 
 |                                                          
/
$$\int \frac{1}{\left(x^{2} + 9\right)^{3}}\, dx = C + \frac{x \left(9 - x^{2}\right)}{648 \left(x^{2} + 9\right)^{2}} + \frac{x}{162 \left(x^{2} + 9\right)} + \frac{\operatorname{atan}{\left(\frac{x}{3} \right)}}{648}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
    ___       
  \/ 3     pi 
- ----- + ----
   7776   3888
$$- \frac{\sqrt{3}}{7776} + \frac{\pi}{3888}$$ 
=
= 
    ___       
  \/ 3     pi 
- ----- + ----
   7776   3888
$$- \frac{\sqrt{3}}{7776} + \frac{\pi}{3888}$$ 
-sqrt(3)/7776 + pi/3888
Respuesta numérica [src] 
0.000585279642439649
0.000585279642439649

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

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