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Resolución de integrales/ exp(-((x-b)^c)/c^2)/A

Integral de exp(-((x-b)^c)/c^2)/A dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1               
  /               
 |                
 |           c    
 |   -(x - b)     
 |   ----------   
 |        2       
 |       c        
 |  e             
 |  ----------- dx
 |       a        
 |                
/                 
0
∫01e(−1)(−b+x)cc2a dx\int\limits_{0}^{1} \frac{e^{\frac{\left(-1\right) \left(- b + x\right)^{c}}{c^{2}}}}{a}\, dx 
Integral(exp((-(x - b)^c)/c^2)/a, (x, 0, 1))
Solución detallada

  1. La integral del producto de una función por una constante es la constante por la integral de esta función:

    ∫e(−1)(−b+x)cc2a dx=∫e(−1)(−b+x)cc2 dxa\int \frac{e^{\frac{\left(-1\right) \left(- b + x\right)^{c}}{c^{2}}}}{a}\, dx = \frac{\int e^{\frac{\left(-1\right) \left(- b + x\right)^{c}}{c^{2}}}\, dx}{a}

    1. No puedo encontrar los pasos en la búsqueda de esta integral.

      Pero la integral

      c2cΓ(1c)γ(1c,(−b+x)cc2)c2Γ(1+1c)\frac{c^{\frac{2}{c}} \Gamma\left(\frac{1}{c}\right) \gamma\left(\frac{1}{c}, \frac{\left(- b + x\right)^{c}}{c^{2}}\right)}{c^{2} \Gamma\left(1 + \frac{1}{c}\right)}

    Por lo tanto, el resultado es: c2cΓ(1c)γ(1c,(−b+x)cc2)ac2Γ(1+1c)\frac{c^{\frac{2}{c}} \Gamma\left(\frac{1}{c}\right) \gamma\left(\frac{1}{c}, \frac{\left(- b + x\right)^{c}}{c^{2}}\right)}{a c^{2} \Gamma\left(1 + \frac{1}{c}\right)}

  2. Ahora simplificar:

    c−1+2cγ(1c,(−b+x)cc2)a\frac{c^{-1 + \frac{2}{c}} \gamma\left(\frac{1}{c}, \frac{\left(- b + x\right)^{c}}{c^{2}}\right)}{a}

  3. Añadimos la constante de integración:

    c−1+2cγ(1c,(−b+x)cc2)a+constant\frac{c^{-1 + \frac{2}{c}} \gamma\left(\frac{1}{c}, \frac{\left(- b + x\right)^{c}}{c^{2}}\right)}{a}+ \mathrm{constant}

Respuesta:

c−1+2cγ(1c,(−b+x)cc2)a+constant\frac{c^{-1 + \frac{2}{c}} \gamma\left(\frac{1}{c}, \frac{\left(- b + x\right)^{c}}{c^{2}}\right)}{a}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                                                        
 |                                                         
 |          c            2                                 
 |  -(x - b)             -                    /          c\
 |  ----------           c      /1\           |1  (x - b) |
 |       2              c *Gamma|-|*lowergamma|-, --------|
 |      c                       \c/           |c      2   |
 | e                                          \      c    /
 | ----------- dx = C + -----------------------------------
 |      a                           2      /    1\         
 |                               a*c *Gamma|1 + -|         
/                                          \    c/
∫e(−1)(−b+x)cc2a dx=C+c2cΓ(1c)γ(1c,(−b+x)cc2)ac2Γ(1+1c)\int \frac{e^{\frac{\left(-1\right) \left(- b + x\right)^{c}}{c^{2}}}}{a}\, dx = C + \frac{c^{\frac{2}{c}} \Gamma\left(\frac{1}{c}\right) \gamma\left(\frac{1}{c}, \frac{\left(- b + x\right)^{c}}{c^{2}}\right)}{a c^{2} \Gamma\left(1 + \frac{1}{c}\right)}
Simplificar
Respuesta [src] 
 2                                     2                              
 -                    /          c\    -                    /       c\
 c      /1\           |1  (1 - b) |    c      /1\           |1  (-b) |
c *Gamma|-|*lowergamma|-, --------|   c *Gamma|-|*lowergamma|-, -----|
        \c/           |c      2   |           \c/           |c     2 |
                      \      c    /                         \     c  /
----------------------------------- - --------------------------------
            2      /    1\                      2      /    1\        
         a*c *Gamma|1 + -|                   a*c *Gamma|1 + -|        
                   \    c/                             \    c/
−c2cΓ(1c)γ(1c,(−b)cc2)ac2Γ(1+1c)+c2cΓ(1c)γ(1c,(1−b)cc2)ac2Γ(1+1c)- \frac{c^{\frac{2}{c}} \Gamma\left(\frac{1}{c}\right) \gamma\left(\frac{1}{c}, \frac{\left(- b\right)^{c}}{c^{2}}\right)}{a c^{2} \Gamma\left(1 + \frac{1}{c}\right)} + \frac{c^{\frac{2}{c}} \Gamma\left(\frac{1}{c}\right) \gamma\left(\frac{1}{c}, \frac{\left(1 - b\right)^{c}}{c^{2}}\right)}{a c^{2} \Gamma\left(1 + \frac{1}{c}\right)} 
=
= 
 2                                     2                              
 -                    /          c\    -                    /       c\
 c      /1\           |1  (1 - b) |    c      /1\           |1  (-b) |
c *Gamma|-|*lowergamma|-, --------|   c *Gamma|-|*lowergamma|-, -----|
        \c/           |c      2   |           \c/           |c     2 |
                      \      c    /                         \     c  /
----------------------------------- - --------------------------------
            2      /    1\                      2      /    1\        
         a*c *Gamma|1 + -|                   a*c *Gamma|1 + -|        
                   \    c/                             \    c/
−c2cΓ(1c)γ(1c,(−b)cc2)ac2Γ(1+1c)+c2cΓ(1c)γ(1c,(1−b)cc2)ac2Γ(1+1c)- \frac{c^{\frac{2}{c}} \Gamma\left(\frac{1}{c}\right) \gamma\left(\frac{1}{c}, \frac{\left(- b\right)^{c}}{c^{2}}\right)}{a c^{2} \Gamma\left(1 + \frac{1}{c}\right)} + \frac{c^{\frac{2}{c}} \Gamma\left(\frac{1}{c}\right) \gamma\left(\frac{1}{c}, \frac{\left(1 - b\right)^{c}}{c^{2}}\right)}{a c^{2} \Gamma\left(1 + \frac{1}{c}\right)} 
c^(2/c)*gamma(1/c)*lowergamma(1/c, (1 - b)^c/c^2)/(a*c^2*gamma(1 + 1/c)) - c^(2/c)*gamma(1/c)*lowergamma(1/c, (-b)^c/c^2)/(a*c^2*gamma(1 + 1/c))

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

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