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Resolución de integrales/ exp(-ax)/ x*exp(-ax)*dx

Integral de x*exp(-ax)*dx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 10           
  /           
 |            
 |     -a*x   
 |  x*e     dx
 |            
/             
0
∫010xe−ax dx\int\limits_{0}^{10} x e^{- a x}\, dx 
Integral(x*exp((-a)*x), (x, 0, 10))
Respuesta (Indefinida) [src] 
                    //         2                    \                          
                    ||        x                     |                          
                    ||        --           for a = 0|                          
                    ||        2                     |                          
  /                 ||                              |     //   x     for a = 0\
 |                  ||/ -a*x                        |     ||                  |
 |    -a*x          |||e           2                |     ||  -a*x            |
 | x*e     dx = C - |<|-----  for a  != 0           | + x*|<-e                |
 |                  |||   2                         |     ||-------  otherwise|
/                   ||<  a                 otherwise|     ||   a              |
                    |||                             |     \\                  /
                    ||| -x                          |                          
                    ||| ---    otherwise            |                          
                    ||\  a                          |                          
                    \\                              /
∫xe−ax dx=C+x({xfor a=0−e−axaotherwise)−{x22for a=0{e−axa2for a2≠0−xaotherwiseotherwise\int x e^{- a x}\, dx = C + x \left(\begin{cases} x & \text{for}\: a = 0 \\- \frac{e^{- a x}}{a} & \text{otherwise} \end{cases}\right) - \begin{cases} \frac{x^{2}}{2} & \text{for}\: a = 0 \\\begin{cases} \frac{e^{- a x}}{a^{2}} & \text{for}\: a^{2} \neq 0 \\- \frac{x}{a} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/                  -10*a                                  
|1    (-1 - 10*a)*e                                       
|-- + ------------------  for And(a > -oo, a < oo, a != 0)
< 2            2                                          
|a            a                                           
|                                                         
\          50                        otherwise
{(−10a−1)e−10aa2+1a2for a>−∞∧a<∞∧a≠050otherwise\begin{cases} \frac{\left(- 10 a - 1\right) e^{- 10 a}}{a^{2}} + \frac{1}{a^{2}} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\50 & \text{otherwise} \end{cases} 
=
= 
/                  -10*a                                  
|1    (-1 - 10*a)*e                                       
|-- + ------------------  for And(a > -oo, a < oo, a != 0)
< 2            2                                          
|a            a                                           
|                                                         
\          50                        otherwise
{(−10a−1)e−10aa2+1a2for a>−∞∧a<∞∧a≠050otherwise\begin{cases} \frac{\left(- 10 a - 1\right) e^{- 10 a}}{a^{2}} + \frac{1}{a^{2}} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\50 & \text{otherwise} \end{cases} 
Piecewise((a^(-2) + (-1 - 10*a)*exp(-10*a)/a^2, (a > -oo)∧(a < oo)∧(Ne(a, 0))), (50, True))

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

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