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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

    TrigSubstitutionRule(theta=_theta, func=tan(_theta), rewritten=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), restriction=True, context=1/((x**2 + 1)**3), symbol=x)

  1. Ahora simplificar:

  2. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
  /                                                       
 |                                                /     2\
 |     1              3*atan(x)       x         x*\1 - x /
 | --------- dx = C + --------- + ---------- + -----------
 |         3              8         /     2\             2
 | / 2    \                       2*\1 + x /     /     2\ 
 | \x  + 1/                                    8*\1 + x / 
 |                                                        
/
$$\int \frac{1}{\left(x^{2} + 1\right)^{3}}\, dx = C + \frac{x \left(1 - x^{2}\right)}{8 \left(x^{2} + 1\right)^{2}} + \frac{x}{2 \left(x^{2} + 1\right)} + \frac{3 \operatorname{atan}{\left(x \right)}}{8}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
1   3*pi
- + ----
4    32
$$\frac{1}{4} + \frac{3 \pi}{32}$$ 
=
= 
1   3*pi
- + ----
4    32
$$\frac{1}{4} + \frac{3 \pi}{32}$$ 
1/4 + 3*pi/32
Respuesta numérica [src] 
0.544524311274043
0.544524311274043

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

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