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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

    TrigSubstitutionRule(theta=_theta, func=tan(_theta), rewritten=cos(_theta)**6, substep=RewriteRule(rewritten=(cos(2*_theta)/2 + 1/2)**3, substep=AlternativeRule(alternatives=[RewriteRule(rewritten=cos(2*_theta)**3/8 + 3*cos(2*_theta)**2/8 + 3*cos(2*_theta)/8 + 1/8, substep=AddRule(substeps=[ConstantTimesRule(constant=1/8, other=cos(2*_theta)**3, substep=RewriteRule(rewritten=(1 - sin(2*_theta)**2)*cos(2*_theta), substep=AlternativeRule(alternatives=[AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1, substep=AddRule(substeps=[ConstantRule(constant=1/2, context=1/2, symbol=_u), ConstantTimesRule(constant=-1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=-_u**2/2, symbol=_u)], context=1/2 - _u**2/2, symbol=_u), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1, substep=AddRule(substeps=[ConstantTimesRule(constant=-1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=-sin(_u)**2*cos(_u)/2, symbol=_u), ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u)/2, symbol=_u)], context=-sin(_u)**2*cos(_u)/2 + cos(_u)/2, symbol=_u), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta)], context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), RewriteRule(rewritten=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(2*_theta)**2*cos(2*_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1/2, substep=ConstantTimesRule(constant=1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta)], context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), context=-sin(2*_theta)**2*cos(2*_theta), symbol=_theta), 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=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), symbol=_theta), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), RewriteRule(rewritten=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(2*_theta)**2*cos(2*_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1/2, substep=ConstantTimesRule(constant=1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta)], context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), context=-sin(2*_theta)**2*cos(2*_theta), symbol=_theta), 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=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), symbol=_theta), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta)], context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), context=cos(2*_theta)**3, symbol=_theta), context=cos(2*_theta)**3/8, symbol=_theta), ConstantTimesRule(constant=3/8, 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=3*cos(2*_theta)**2/8, symbol=_theta), ConstantTimesRule(constant=3/8, 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=3*cos(2*_theta)/8, symbol=_theta), ConstantRule(constant=1/8, context=1/8, symbol=_theta)], context=cos(2*_theta)**3/8 + 3*cos(2*_theta)**2/8 + 3*cos(2*_theta)/8 + 1/8, symbol=_theta), context=(cos(2*_theta)/2 + 1/2)**3, symbol=_theta), RewriteRule(rewritten=cos(2*_theta)**3/8 + 3*cos(2*_theta)**2/8 + 3*cos(2*_theta)/8 + 1/8, substep=AddRule(substeps=[ConstantTimesRule(constant=1/8, other=cos(2*_theta)**3, substep=RewriteRule(rewritten=(1 - sin(2*_theta)**2)*cos(2*_theta), substep=AlternativeRule(alternatives=[AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1, substep=AddRule(substeps=[ConstantRule(constant=1/2, context=1/2, symbol=_u), ConstantTimesRule(constant=-1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=-_u**2/2, symbol=_u)], context=1/2 - _u**2/2, symbol=_u), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1, substep=AddRule(substeps=[ConstantTimesRule(constant=-1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=-sin(_u)**2*cos(_u)/2, symbol=_u), ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u)/2, symbol=_u)], context=-sin(_u)**2*cos(_u)/2 + cos(_u)/2, symbol=_u), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta)], context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), RewriteRule(rewritten=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(2*_theta)**2*cos(2*_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1/2, substep=ConstantTimesRule(constant=1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta)], context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), context=-sin(2*_theta)**2*cos(2*_theta), symbol=_theta), 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=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), symbol=_theta), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), RewriteRule(rewritten=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(2*_theta)**2*cos(2*_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1/2, substep=ConstantTimesRule(constant=1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta)], context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), context=-sin(2*_theta)**2*cos(2*_theta), symbol=_theta), 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=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), symbol=_theta), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta)], context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), context=cos(2*_theta)**3, symbol=_theta), context=cos(2*_theta)**3/8, symbol=_theta), ConstantTimesRule(constant=3/8, 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=3*cos(2*_theta)**2/8, symbol=_theta), ConstantTimesRule(constant=3/8, 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=3*cos(2*_theta)/8, symbol=_theta), ConstantRule(constant=1/8, context=1/8, symbol=_theta)], context=cos(2*_theta)**3/8 + 3*cos(2*_theta)**2/8 + 3*cos(2*_theta)/8 + 1/8, symbol=_theta), context=(cos(2*_theta)/2 + 1/2)**3, symbol=_theta)], context=(cos(2*_theta)/2 + 1/2)**3, symbol=_theta), context=cos(_theta)**6, symbol=_theta), restriction=True, context=1/((x**2 + 1)**4), symbol=x)

  1. Ahora simplificar:

  2. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
  /                                                                      
 |                                                   3           /     2\
 |     1              5*atan(x)       x             x        3*x*\1 - x /
 | --------- dx = C + --------- + ---------- - ----------- + ------------
 |         4              16        /     2\             3              2
 | / 2    \                       2*\1 + x /     /     2\       /     2\ 
 | \x  + 1/                                    6*\1 + x /    16*\1 + x / 
 |                                                                       
/
$$\int \frac{1}{\left(x^{2} + 1\right)^{4}}\, dx = C - \frac{x^{3}}{6 \left(x^{2} + 1\right)^{3}} + \frac{3 x \left(1 - x^{2}\right)}{16 \left(x^{2} + 1\right)^{2}} + \frac{x}{2 \left(x^{2} + 1\right)} + \frac{5 \operatorname{atan}{\left(x \right)}}{16}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
11   5*pi
-- + ----
48    64
$$\frac{11}{48} + \frac{5 \pi}{64}$$ 
=
= 
11   5*pi
-- + ----
48    64
$$\frac{11}{48} + \frac{5 \pi}{64}$$ 
11/48 + 5*pi/64
Respuesta numérica [src] 
0.474603592728369
0.474603592728369

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

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