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Resolución de integrales/ 2*t^2/ k^2*t^2*exp(t*(-k))/3

Integral de k^2*t^2*exp(t*(-k))/3 dt

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo                 
  /                 
 |                  
 |   2  2  t*(-k)   
 |  k *t *e         
 |  ------------- dt
 |        3         
 |                  
/                   
0
∫0∞k2t2e−kt3 dt\int\limits_{0}^{\infty} \frac{k^{2} t^{2} e^{- k t}}{3}\, dt 
Integral(((k^2*t^2)*exp(t*(-k)))/3, (t, 0, oo))
Respuesta (Indefinida) [src] 
                             /    //              3                         \                           \
                             |    ||             t                          |                           |
                             |    ||             --                for k = 0|                           |
                             |    ||             3                          |                           |
                             |    ||                                        |                           |
                             |    ||/           -k*t                        |      //   t     for k = 0\|
                             |    |||(1 + k*t)*e           3                |      ||                  ||
                           2 |    |||---------------  for k  != 0           |    2 ||  -k*t            ||
                          k *|- 2*|<|        3                              | + t *|<-e                ||
                             |    |||       k                               |      ||-------  otherwise||
                             |    ||<                              otherwise|      ||   k              ||
                             |    |||       2                               |      \\                  /|
                             |    |||     -t                                |                           |
  /                          |    |||     ----         otherwise            |                           |
 |                           |    |||     2*k                               |                           |
 |  2  2  t*(-k)             |    ||\                                       |                           |
 | k *t *e                   \    \\                                        /                           /
 | ------------- dt = C + -------------------------------------------------------------------------------
 |       3                                                       3                                       
 |                                                                                                       
/
∫k2t2e−kt3 dt=C+k2(t2({tfor k=0−e−ktkotherwise)−2({t33for k=0{(kt+1)e−ktk3for k3≠0−t22kotherwiseotherwise))3\int \frac{k^{2} t^{2} e^{- k t}}{3}\, dt = C + \frac{k^{2} \left(t^{2} \left(\begin{cases} t & \text{for}\: k = 0 \\- \frac{e^{- k t}}{k} & \text{otherwise} \end{cases}\right) - 2 \left(\begin{cases} \frac{t^{3}}{3} & \text{for}\: k = 0 \\\begin{cases} \frac{\left(k t + 1\right) e^{- k t}}{k^{3}} & \text{for}\: k^{3} \neq 0 \\- \frac{t^{2}}{2 k} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)}{3}
Simplificar
Respuesta [src] 
/        2                          pi
|       ---          for |arg(k)| < --
|       3*k                         2 
|                                     
| oo                                  
|  /                                  
| |                                   
< |   2  2  -k*t                      
| |  k *t *e                          
| |  ----------- dt      otherwise    
| |       3                           
| |                                   
|/                                    
|0                                    
\
{23kfor ∣arg⁡(k)∣<π2∫0∞k2t2e−kt3 dtotherwise\begin{cases} \frac{2}{3 k} & \text{for}\: \left|{\arg{\left(k \right)}}\right| < \frac{\pi}{2} \\\int\limits_{0}^{\infty} \frac{k^{2} t^{2} e^{- k t}}{3}\, dt & \text{otherwise} \end{cases} 
=
= 
/        2                          pi
|       ---          for |arg(k)| < --
|       3*k                         2 
|                                     
| oo                                  
|  /                                  
| |                                   
< |   2  2  -k*t                      
| |  k *t *e                          
| |  ----------- dt      otherwise    
| |       3                           
| |                                   
|/                                    
|0                                    
\
{23kfor ∣arg⁡(k)∣<π2∫0∞k2t2e−kt3 dtotherwise\begin{cases} \frac{2}{3 k} & \text{for}\: \left|{\arg{\left(k \right)}}\right| < \frac{\pi}{2} \\\int\limits_{0}^{\infty} \frac{k^{2} t^{2} e^{- k t}}{3}\, dt & \text{otherwise} \end{cases} 
Piecewise((2/(3*k), Abs(arg(k)) < pi/2), (Integral(k^2*t^2*exp(-k*t)/3, (t, 0, oo)), True))

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

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