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Resolución de integrales/ (t-1)/ x*e^(x*(t-1))

Integral de x*e^(x*(t-1)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo                
  /                
 |                 
 |     x*(t - 1)   
 |  x*E          dx
 |                 
/                  
0
∫0∞ex(t−1)x dx\int\limits_{0}^{\infty} e^{x \left(t - 1\right)} x\, dx 
Integral(x*E^(x*(t - 1)), (x, 0, oo))
Respuesta (Indefinida) [src] 
                         //                  2                            \                                   
                         ||                 x                             |                                   
                         ||                 --                   for t = 1|                                   
                         ||                 2                             |                                   
  /                      ||                                               |     //     x       for -1 + t = 0\
 |                       ||/ x*(-1 + t)                                   |     ||                           |
 |    x*(t - 1)          |||e                      2                      |     || x*(-1 + t)                |
 | x*E          dx = C - |<|------------  for 1 + t  - 2*t != 0           | + x*|
∫ex(t−1)x dx=C+x({xfor t−1=0ex(t−1)t−1otherwise)−{x22for t=1{ex(t−1)t2−2t+1for t2−2t+1≠0xt−1otherwiseotherwise\int e^{x \left(t - 1\right)} x\, dx = C + x \left(\begin{cases} x & \text{for}\: t - 1 = 0 \\\frac{e^{x \left(t - 1\right)}}{t - 1} & \text{otherwise} \end{cases}\right) - \begin{cases} \frac{x^{2}}{2} & \text{for}\: t = 1 \\\begin{cases} \frac{e^{x \left(t - 1\right)}}{t^{2} - 2 t + 1} & \text{for}\: t^{2} - 2 t + 1 \neq 0 \\\frac{x}{t - 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/         1                                     pi
|     ---------        for |pi + arg(-1 + t)| < --
|             2                                 2 
|     (-1 + t)                                    
|                                                 
| oo                                              
<  /                                              
| |                                               
| |     x*(-1 + t)                                
| |  x*e           dx           otherwise         
| |                                               
|/                                                
\0
{1(t−1)2for ∣arg⁡(t−1)+π∣<π2∫0∞xex(t−1) dxotherwise\begin{cases} \frac{1}{\left(t - 1\right)^{2}} & \text{for}\: \left|{\arg{\left(t - 1 \right)} + \pi}\right| < \frac{\pi}{2} \\\int\limits_{0}^{\infty} x e^{x \left(t - 1\right)}\, dx & \text{otherwise} \end{cases} 
=
= 
/         1                                     pi
|     ---------        for |pi + arg(-1 + t)| < --
|             2                                 2 
|     (-1 + t)                                    
|                                                 
| oo                                              
<  /                                              
| |                                               
| |     x*(-1 + t)                                
| |  x*e           dx           otherwise         
| |                                               
|/                                                
\0
{1(t−1)2for ∣arg⁡(t−1)+π∣<π2∫0∞xex(t−1) dxotherwise\begin{cases} \frac{1}{\left(t - 1\right)^{2}} & \text{for}\: \left|{\arg{\left(t - 1 \right)} + \pi}\right| < \frac{\pi}{2} \\\int\limits_{0}^{\infty} x e^{x \left(t - 1\right)}\, dx & \text{otherwise} \end{cases} 
Piecewise(((-1 + t)^(-2), Abs(pi + arg(-1 + t)) < pi/2), (Integral(x*exp(x*(-1 + t)), (x, 0, oo)), True))

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

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