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Resolución de integrales/ 1/(5+x^2)/ 1/(5+x^2)^3

Integral de 1/(5+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\    
 |  \5 + x /    
 |              
/               
0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} + 5\right)^{3}}\, dx$$ 
Integral(1/((5 + x^2)^3), (x, 0, 1))
Solución detallada

    TrigSubstitutionRule(theta=_theta, func=sqrt(5)*tan(_theta), rewritten=sqrt(5)*cos(_theta)**4/125, substep=ConstantTimesRule(constant=sqrt(5)/125, 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=sqrt(5)*cos(_theta)**4/125, symbol=_theta), restriction=True, context=1/((x**2 + 5)**3), symbol=x)

  1. Ahora simplificar:

  2. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
                            /      /    ___\                                \
                            |      |x*\/ 5 |                                |
                            |3*atan|-------|        ___         ___ /     2\|
                        ___ |      \   5   /    x*\/ 5      x*\/ 5 *\5 - x /|
                      \/ 5 *|--------------- + ---------- + ----------------|
  /                         |       8            /     2\               2   |
 |                          |                  2*\5 + x /       /     2\    |
 |     1                    \                                 8*\5 + x /    /
 | --------- dx = C + -------------------------------------------------------
 |         3                                    125                          
 | /     2\                                                                  
 | \5 + x /                                                                  
 |                                                                           
/
$$\int \frac{1}{\left(x^{2} + 5\right)^{3}}\, dx = C + \frac{\sqrt{5} \left(\frac{\sqrt{5} x \left(5 - x^{2}\right)}{8 \left(x^{2} + 5\right)^{2}} + \frac{\sqrt{5} x}{2 \left(x^{2} + 5\right)} + \frac{3 \operatorname{atan}{\left(\frac{\sqrt{5} x}{5} \right)}}{8}\right)}{125}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                   /  ___\
           ___     |\/ 5 |
       3*\/ 5 *atan|-----|
 7                 \  5  /
---- + -------------------
1800           1000
$$\frac{3 \sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5}}{5} \right)}}{1000} + \frac{7}{1800}$$ 
=
= 
                   /  ___\
           ___     |\/ 5 |
       3*\/ 5 *atan|-----|
 7                 \  5  /
---- + -------------------
1800           1000
$$\frac{3 \sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5}}{5} \right)}}{1000} + \frac{7}{1800}$$ 
7/1800 + 3*sqrt(5)*atan(sqrt(5)/5)/1000
Respuesta numérica [src] 
0.00670991897059179
0.00670991897059179

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

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