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Resolución de integrales/ -a*x*exp(-ax-by)

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo                    
  /                    
 |                     
 |        -a*x - b*y   
 |  -a*x*e           dx
 |                     
/                      
0
∫0∞−axe−ax−by dx\int\limits_{0}^{\infty} - a x e^{- a x - b y}\, dx 
Integral(((-a)*x)*exp((-a)*x - b*y), (x, 0, oo))
Respuesta (Indefinida) [src] 
                               /  //         2                    \                          \      
                               |  ||        x                     |                          |      
                               |  ||        --           for a = 0|                          |      
                               |  ||        2                     |                          |      
  /                            |  ||                              |     //   x     for a = 0\|      
 |                             |  ||/ -a*x                        |     ||                  ||      
 |       -a*x - b*y            |  |||e           2                |     ||  -a*x            ||  -b*y
 | -a*x*e           dx = C - a*|- |<|-----  for a  != 0           | + x*|<-e                ||*e    
 |                             |  |||   2                         |     ||-------  otherwise||      
/                              |  ||<  a                 otherwise|     ||   a              ||      
                               |  |||                             |     \\                  /|      
                               |  ||| -x                          |                          |      
                               |  ||| ---    otherwise            |                          |      
                               |  ||\  a                          |                          |      
                               \  \\                              /                          /
∫−axe−ax−by dx=C−a(x({xfor a=0−e−axaotherwise)−{x22for a=0{e−axa2for a2≠0−xaotherwiseotherwise)e−by\int - a x e^{- a x - b y}\, dx = C - a \left(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}\right) e^{- b y}
Simplificar
Respuesta [src] 
/          -b*y                            
|        -e                              pi
|        -------          for |arg(a)| < --
|           a                            2 
|                                          
| oo                                       
<  /                                       
| |                                        
| |        -a*x - b*y                      
| |  -a*x*e           dx      otherwise    
| |                                        
|/                                         
\0
{−e−byafor ∣arg⁡(a)∣<π2∫0∞(−axe−ax−by) dxotherwise\begin{cases} - \frac{e^{- b y}}{a} & \text{for}\: \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2} \\\int\limits_{0}^{\infty} \left(- a x e^{- a x - b y}\right)\, dx & \text{otherwise} \end{cases} 
=
= 
/          -b*y                            
|        -e                              pi
|        -------          for |arg(a)| < --
|           a                            2 
|                                          
| oo                                       
<  /                                       
| |                                        
| |        -a*x - b*y                      
| |  -a*x*e           dx      otherwise    
| |                                        
|/                                         
\0
{−e−byafor ∣arg⁡(a)∣<π2∫0∞(−axe−ax−by) dxotherwise\begin{cases} - \frac{e^{- b y}}{a} & \text{for}\: \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2} \\\int\limits_{0}^{\infty} \left(- a x e^{- a x - b y}\right)\, dx & \text{otherwise} \end{cases} 
Piecewise((-exp(-b*y)/a, Abs(arg(a)) < pi/2), (Integral(-a*x*exp(-a*x - b*y), (x, 0, oo)), True))

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

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