Sr Examen

Otras calculadoras

Resolución de integrales/ exp(t)/ t*exp(t)*exp(-p*t)

Integral de t*exp(t)*exp(-p*t) dt

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo              
  /              
 |               
 |     t  -p*t   
 |  t*e *e     dt
 |               
/                
0
$$\int\limits_{0}^{\infty} t e^{t} e^{- p t}\, dt$$ 
Integral((t*exp(t))*exp((-p)*t), (t, 0, oo))
Respuesta (Indefinida) [src] 
                       //                  2                            \                                    
                       ||                 t                             |                                    
                       ||                 --                   for p = 1|                                    
                       ||                 2                             |                                    
  /                    ||                                               |     //     t        for -1 + p = 0\
 |                     ||/  t*(1 - p)                                   |     ||                            |
 |    t  -p*t          ||| e                     2                      |     ||  t*(1 - p)                 |
 | t*e *e     dt = C - |<|------------  for 1 + p  - 2*p != 0           | + t*|<-e                          |
 |                     |||     2                                        |     ||------------    otherwise   |
/                      ||<1 + p  - 2*p                         otherwise|     ||   -1 + p                   |
                       |||                                              |     \\                            /
                       |||    -t                                        |                                    
                       |||   ------           otherwise                 |                                    
                       ||\   -1 + p                                     |                                    
                       \\                                               /
$$\int t e^{t} e^{- p t}\, dt = C + t \left(\begin{cases} t & \text{for}\: p - 1 = 0 \\- \frac{e^{t \left(1 - p\right)}}{p - 1} & \text{otherwise} \end{cases}\right) - \begin{cases} \frac{t^{2}}{2} & \text{for}\: p = 1 \\\begin{cases} \frac{e^{t \left(1 - p\right)}}{p^{2} - 2 p + 1} & \text{for}\: p^{2} - 2 p + 1 \neq 0 \\- \frac{t}{p - 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$
Simplificar
Respuesta [src] 
/       1                  /                                       pi  1     \
|    --------       for And|-1 + |p|*cos(|arg(p)|) > 0, |arg(p)| < --, - != 1|
|           2              \                                       2   p     /
|    (1 - p)                                                                  
|                                                                             
| oo                                                                          
<  /                                                                          
| |                                                                           
| |     t  -p*t                                                               
| |  t*e *e     dt                          otherwise                         
| |                                                                           
|/                                                                            
\0
$$\begin{cases} \frac{1}{\left(1 - p\right)^{2}} & \text{for}\: \cos{\left(\left|{\arg{\left(p \right)}}\right| \right)} \left|{p}\right| - 1 > 0 \wedge \left|{\arg{\left(p \right)}}\right| < \frac{\pi}{2} \wedge \frac{1}{p} \neq 1 \\\int\limits_{0}^{\infty} t e^{t} e^{- p t}\, dt & \text{otherwise} \end{cases}$$ 
=
= 
/       1                  /                                       pi  1     \
|    --------       for And|-1 + |p|*cos(|arg(p)|) > 0, |arg(p)| < --, - != 1|
|           2              \                                       2   p     /
|    (1 - p)                                                                  
|                                                                             
| oo                                                                          
<  /                                                                          
| |                                                                           
| |     t  -p*t                                                               
| |  t*e *e     dt                          otherwise                         
| |                                                                           
|/                                                                            
\0
$$\begin{cases} \frac{1}{\left(1 - p\right)^{2}} & \text{for}\: \cos{\left(\left|{\arg{\left(p \right)}}\right| \right)} \left|{p}\right| - 1 > 0 \wedge \left|{\arg{\left(p \right)}}\right| < \frac{\pi}{2} \wedge \frac{1}{p} \neq 1 \\\int\limits_{0}^{\infty} t e^{t} e^{- p t}\, dt & \text{otherwise} \end{cases}$$ 
Piecewise(((1 - p)^(-2), (Ne(1/p, 1))∧(Abs(arg(p)) < pi/2)∧(-1 + |p|*cos(Abs(arg(p))) > 0)), (Integral(t*exp(t)*exp(-p*t), (t, 0, oo)), True))

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

    © Sr Examen