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Resolución de integrales/ cos(t)/ e^(-st)/ t*cos(t)*e^(-st)

Integral de t*cos(t)*e^(-st) dt

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo                  
  /                  
 |                   
 |            -s*t   
 |  t*cos(t)*E     dt
 |                   
/                    
0
∫0∞e−sttcos⁡(t) dt\int\limits_{0}^{\infty} e^{- s t} t \cos{\left(t \right)}\, dt 
Integral((t*cos(t))*E^((-s)*t), (t, 0, oo))
Respuesta (Indefinida) [src] 
                           //                                                                   I*t      I*t           2         I*t               I*t      2  I*t                                                                               \
                           ||                                                           cos(t)*e      t*e   *sin(t)   t *cos(t)*e      I*t*cos(t)*e      I*t *e   *sin(t)                                                                        |
                           ||                                                           ----------- + ------------- + -------------- - --------------- - ----------------                                                              for s = -I|
                           ||                                                                4              4               4                 4                 4                                                                                |
  /                        ||                                                                                                                                                                                                                    |
 |                         ||                                                                 -I*t      -I*t           2         -I*t               -I*t      2  -I*t                                                                            |
 |           -s*t          ||                                                         cos(t)*e       t*e    *sin(t)   t *cos(t)*e       I*t*cos(t)*e       I*t *e    *sin(t)                                                                     |
 | t*cos(t)*E     dt = C + |<                                                         ------------ + -------------- + --------------- + ---------------- + -----------------                                                           for s = I |
 |                         ||                                                              4               4                 4                 4                   4                                                                             |
/                          ||                                                                                                                                                                                                                    |
                           ||                                                                   2                                                          2                                                         3                           |
                           ||          cos(t)                      t*sin(t)                    s *cos(t)                    2*s*sin(t)                  t*s *sin(t)                   s*t*cos(t)                  t*s *cos(t)                    |
                           ||-------------------------- + -------------------------- - -------------------------- + -------------------------- + -------------------------- - -------------------------- - --------------------------  otherwise |
                           || 4  s*t      2  s*t    s*t    4  s*t      2  s*t    s*t    4  s*t      2  s*t    s*t    4  s*t      2  s*t    s*t    4  s*t      2  s*t    s*t    4  s*t      2  s*t    s*t    4  s*t      2  s*t    s*t            |
                           \\s *e    + 2*s *e    + e      s *e    + 2*s *e    + e      s *e    + 2*s *e    + e      s *e    + 2*s *e    + e      s *e    + 2*s *e    + e      s *e    + 2*s *e    + e      s *e    + 2*s *e    + e               /
∫e−sttcos⁡(t) dt=C+{−it2eitsin⁡(t)4+t2eitcos⁡(t)4+teitsin⁡(t)4−iteitcos⁡(t)4+eitcos⁡(t)4for s=−iit2e−itsin⁡(t)4+t2e−itcos⁡(t)4+te−itsin⁡(t)4+ite−itcos⁡(t)4+e−itcos⁡(t)4for s=i−s3tcos⁡(t)s4est+2s2est+est+s2tsin⁡(t)s4est+2s2est+est−s2cos⁡(t)s4est+2s2est+est−stcos⁡(t)s4est+2s2est+est+2ssin⁡(t)s4est+2s2est+est+tsin⁡(t)s4est+2s2est+est+cos⁡(t)s4est+2s2est+estotherwise\int e^{- s t} t \cos{\left(t \right)}\, dt = C + \begin{cases} - \frac{i t^{2} e^{i t} \sin{\left(t \right)}}{4} + \frac{t^{2} e^{i t} \cos{\left(t \right)}}{4} + \frac{t e^{i t} \sin{\left(t \right)}}{4} - \frac{i t e^{i t} \cos{\left(t \right)}}{4} + \frac{e^{i t} \cos{\left(t \right)}}{4} & \text{for}\: s = - i \\\frac{i t^{2} e^{- i t} \sin{\left(t \right)}}{4} + \frac{t^{2} e^{- i t} \cos{\left(t \right)}}{4} + \frac{t e^{- i t} \sin{\left(t \right)}}{4} + \frac{i t e^{- i t} \cos{\left(t \right)}}{4} + \frac{e^{- i t} \cos{\left(t \right)}}{4} & \text{for}\: s = i \\- \frac{s^{3} t \cos{\left(t \right)}}{s^{4} e^{s t} + 2 s^{2} e^{s t} + e^{s t}} + \frac{s^{2} t \sin{\left(t \right)}}{s^{4} e^{s t} + 2 s^{2} e^{s t} + e^{s t}} - \frac{s^{2} \cos{\left(t \right)}}{s^{4} e^{s t} + 2 s^{2} e^{s t} + e^{s t}} - \frac{s t \cos{\left(t \right)}}{s^{4} e^{s t} + 2 s^{2} e^{s t} + e^{s t}} + \frac{2 s \sin{\left(t \right)}}{s^{4} e^{s t} + 2 s^{2} e^{s t} + e^{s t}} + \frac{t \sin{\left(t \right)}}{s^{4} e^{s t} + 2 s^{2} e^{s t} + e^{s t}} + \frac{\cos{\left(t \right)}}{s^{4} e^{s t} + 2 s^{2} e^{s t} + e^{s t}} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/       /     2\                           
|     2*\1 - s /                           
|--------------------   for 2*|arg(s)| < pi
|/     2\ /        2\                      
|\1 + s /*\-2 - 2*s /                      
|                                          
| oo                                       
<  /                                       
| |                                        
| |            -s*t                        
| |  t*cos(t)*e     dt       otherwise     
| |                                        
|/                                         
|0                                         
\
{2(1−s2)(−2s2−2)(s2+1)for 2∣arg⁡(s)∣<π∫0∞te−stcos⁡(t) dtotherwise\begin{cases} \frac{2 \left(1 - s^{2}\right)}{\left(- 2 s^{2} - 2\right) \left(s^{2} + 1\right)} & \text{for}\: 2 \left|{\arg{\left(s \right)}}\right| < \pi \\\int\limits_{0}^{\infty} t e^{- s t} \cos{\left(t \right)}\, dt & \text{otherwise} \end{cases} 
=
= 
/       /     2\                           
|     2*\1 - s /                           
|--------------------   for 2*|arg(s)| < pi
|/     2\ /        2\                      
|\1 + s /*\-2 - 2*s /                      
|                                          
| oo                                       
<  /                                       
| |                                        
| |            -s*t                        
| |  t*cos(t)*e     dt       otherwise     
| |                                        
|/                                         
|0                                         
\
{2(1−s2)(−2s2−2)(s2+1)for 2∣arg⁡(s)∣<π∫0∞te−stcos⁡(t) dtotherwise\begin{cases} \frac{2 \left(1 - s^{2}\right)}{\left(- 2 s^{2} - 2\right) \left(s^{2} + 1\right)} & \text{for}\: 2 \left|{\arg{\left(s \right)}}\right| < \pi \\\int\limits_{0}^{\infty} t e^{- s t} \cos{\left(t \right)}\, dt & \text{otherwise} \end{cases} 
Piecewise((2*(1 - s^2)/((1 + s^2)*(-2 - 2*s^2)), 2*Abs(arg(s)) < pi), (Integral(t*cos(t)*exp(-s*t), (t, 0, oo)), True))

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

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