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Resolución de integrales/ 2*t^2/ t^2/sqrt(1-2*t^2)

Integral de t^2/sqrt(1-2*t^2) dt

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
    ___                 
  \/ 2                  
  -----                 
    2                   
    /                   
   |                    
   |           2        
   |          t         
   |    ------------- dt
   |       __________   
   |      /        2    
   |    \/  1 - 2*t     
   |                    
  /                     
   ___                  
-\/ 2                   
-------                 
   2
∫−2222t21−2t2 dt\int\limits_{- \frac{\sqrt{2}}{2}}^{\frac{\sqrt{2}}{2}} \frac{t^{2}}{\sqrt{1 - 2 t^{2}}}\, dt 
Integral(t^2/sqrt(1 - 2*t^2), (t, -sqrt(2)/2, sqrt(2)/2))
Solución detallada

    TrigSubstitutionRule(theta=_theta, func=sqrt(2)*sin(_theta)/2, rewritten=sqrt(2)*sin(_theta)**2/4, substep=ConstantTimesRule(constant=sqrt(2)/4, other=sin(_theta)**2, substep=RewriteRule(rewritten=1/2 - cos(2*_theta)/2, substep=AddRule(substeps=[ConstantRule(constant=1/2, context=1/2, 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)], context=1/2 - cos(2*_theta)/2, symbol=_theta), context=sin(_theta)**2, symbol=_theta), context=sqrt(2)*sin(_theta)**2/4, symbol=_theta), restriction=(t > -sqrt(2)/2) & (t < sqrt(2)/2), context=t**2/sqrt(1 - 2*t**2), symbol=t)

  1. Ahora simplificar:

    {−t1−2t24+2asin⁡(2t)8for t>−22∧t<22\begin{cases} - \frac{t \sqrt{1 - 2 t^{2}}}{4} + \frac{\sqrt{2} \operatorname{asin}{\left(\sqrt{2} t \right)}}{8} & \text{for}\: t > - \frac{\sqrt{2}}{2} \wedge t < \frac{\sqrt{2}}{2} \end{cases}

  2. Añadimos la constante de integración:

    {−t1−2t24+2asin⁡(2t)8for t>−22∧t<22+constant\begin{cases} - \frac{t \sqrt{1 - 2 t^{2}}}{4} + \frac{\sqrt{2} \operatorname{asin}{\left(\sqrt{2} t \right)}}{8} & \text{for}\: t > - \frac{\sqrt{2}}{2} \wedge t < \frac{\sqrt{2}}{2} \end{cases}+ \mathrm{constant}

Respuesta:

{−t1−2t24+2asin⁡(2t)8for t>−22∧t<22+constant\begin{cases} - \frac{t \sqrt{1 - 2 t^{2}}}{4} + \frac{\sqrt{2} \operatorname{asin}{\left(\sqrt{2} t \right)}}{8} & \text{for}\: t > - \frac{\sqrt{2}}{2} \wedge t < \frac{\sqrt{2}}{2} \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                                                                                                        
 |                        //      /                           __________\                                 \
 |        2               ||      |    /    ___\       ___   /        2 |                                 |
 |       t                ||  ___ |asin\t*\/ 2 /   t*\/ 2 *\/  1 - 2*t  |                                 |
 | ------------- dt = C + |<\/ 2 *|------------- - ---------------------|         /       ___         ___\|
 |    __________          ||      \      2                   2          /         |    -\/ 2        \/ 2 ||
 |   /        2           ||---------------------------------------------  for And|t > -------, t < -----||
 | \/  1 - 2*t            \\                      4                               \       2           2  //
 |                                                                                                         
/
∫t21−2t2 dt=C+{2(−2t1−2t22+asin⁡(2t)2)4for t>−22∧t<22\int \frac{t^{2}}{\sqrt{1 - 2 t^{2}}}\, dt = C + \begin{cases} \frac{\sqrt{2} \left(- \frac{\sqrt{2} t \sqrt{1 - 2 t^{2}}}{2} + \frac{\operatorname{asin}{\left(\sqrt{2} t \right)}}{2}\right)}{4} & \text{for}\: t > - \frac{\sqrt{2}}{2} \wedge t < \frac{\sqrt{2}}{2} \end{cases}
Simplificar
Gráfica
-0.7-0.6-0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.50.60.7-2525
Trazar el gráfico f(x)
Respuesta [src] 
     ___
pi*\/ 2 
--------
   8
2π8\frac{\sqrt{2} \pi}{8} 
=
= 
     ___
pi*\/ 2 
--------
   8
2π8\frac{\sqrt{2} \pi}{8} 
pi*sqrt(2)/8
Respuesta numérica [src] 
(0.555360363022156 - 1.18612748134533e-8j)
(0.555360363022156 - 1.18612748134533e-8j)

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

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