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Resolución de integrales/ e^(xy/2)/ ye^(xy/2)

Integral de ye^(xy/2) dy

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 log(3)         
    /           
   |            
   |      x*y   
   |      ---   
   |       2    
   |   y*E    dy
   |            
  /             
log(2)
∫log⁡(2)log⁡(3)exy2y dy\int\limits_{\log{\left(2 \right)}}^{\log{\left(3 \right)}} e^{\frac{x y}{2}} y\, dy 
Integral(y*E^((x*y)/2), (y, log(2), log(3)))
Respuesta (Indefinida) [src] 
                   //              x*y             \
  /                ||              ---             |
 |                 ||               2              |
 |    x*y          ||(-4 + 2*x*y)*e          2     |
 |    ---          ||-----------------  for x  != 0|
 |     2           ||         2                    |
 | y*E    dy = C + |<        x                     |
 |                 ||                              |
/                  ||        2                     |
                   ||       y                      |
                   ||       --           otherwise |
                   ||       2                      |
                   \\                              /
∫exy2y dy=C+{(2xy−4)exy2x2for x2≠0y22otherwise\int e^{\frac{x y}{2}} y\, dy = C + \begin{cases} \frac{\left(2 x y - 4\right) e^{\frac{x y}{2}}}{x^{2}} & \text{for}\: x^{2} \neq 0 \\\frac{y^{2}}{2} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/                   x*log(3)                      x*log(2)                                  
|                   --------                      --------                                  
|                      2                             2                                      
|(-4 + 2*x*log(3))*e           (-4 + 2*x*log(2))*e                                          
|--------------------------- - ---------------------------  for And(x > -oo, x < oo, x != 0)
|              2                             2                                              
<             x                             x                                               
|                                                                                           
|                       2         2                                                         
|                    log (3)   log (2)                                                      
|                    ------- - -------                                 otherwise            
|                       2         2                                                         
\
{−(2xlog⁡(2)−4)exlog⁡(2)2x2+(2xlog⁡(3)−4)exlog⁡(3)2x2for x>−∞∧x<∞∧x≠0−log⁡(2)22+log⁡(3)22otherwise\begin{cases} - \frac{\left(2 x \log{\left(2 \right)} - 4\right) e^{\frac{x \log{\left(2 \right)}}{2}}}{x^{2}} + \frac{\left(2 x \log{\left(3 \right)} - 4\right) e^{\frac{x \log{\left(3 \right)}}{2}}}{x^{2}} & \text{for}\: x > -\infty \wedge x < \infty \wedge x \neq 0 \\- \frac{\log{\left(2 \right)}^{2}}{2} + \frac{\log{\left(3 \right)}^{2}}{2} & \text{otherwise} \end{cases} 
=
= 
/                   x*log(3)                      x*log(2)                                  
|                   --------                      --------                                  
|                      2                             2                                      
|(-4 + 2*x*log(3))*e           (-4 + 2*x*log(2))*e                                          
|--------------------------- - ---------------------------  for And(x > -oo, x < oo, x != 0)
|              2                             2                                              
<             x                             x                                               
|                                                                                           
|                       2         2                                                         
|                    log (3)   log (2)                                                      
|                    ------- - -------                                 otherwise            
|                       2         2                                                         
\
{−(2xlog⁡(2)−4)exlog⁡(2)2x2+(2xlog⁡(3)−4)exlog⁡(3)2x2for x>−∞∧x<∞∧x≠0−log⁡(2)22+log⁡(3)22otherwise\begin{cases} - \frac{\left(2 x \log{\left(2 \right)} - 4\right) e^{\frac{x \log{\left(2 \right)}}{2}}}{x^{2}} + \frac{\left(2 x \log{\left(3 \right)} - 4\right) e^{\frac{x \log{\left(3 \right)}}{2}}}{x^{2}} & \text{for}\: x > -\infty \wedge x < \infty \wedge x \neq 0 \\- \frac{\log{\left(2 \right)}^{2}}{2} + \frac{\log{\left(3 \right)}^{2}}{2} & \text{otherwise} \end{cases} 
Piecewise(((-4 + 2*x*log(3))*exp(x*log(3)/2)/x^2 - (-4 + 2*x*log(2))*exp(x*log(2)/2)/x^2, (x > -oo)∧(x < oo)∧(Ne(x, 0))), (log(3)^2/2 - log(2)^2/2, True))

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

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