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Resolución de integrales/ (x-y)/ e^(-(x-y)^2/(4t)-|y|)

Integral de e^(-(x-y)^2/(4t)-|y|) dy

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo                     
  /                     
 |                      
 |           2          
 |   -(x - y)           
 |   ---------- - |y|   
 |      4*t             
 |  E                 dy
 |                      
/                       
-oo
∫−∞∞e−∣y∣+(−1)(x−y)24t dy\int\limits_{-\infty}^{\infty} e^{- \left|{y}\right| + \frac{\left(-1\right) \left(x - y\right)^{2}}{4 t}}\, dy 
Integral(E^((-(x - y)^2)/((4*t)) - |y|), (y, -oo, oo))
Solución detallada

  1. Vuelva a escribir el integrando:

    e−∣y∣+(−1)(x−y)24t=e−x24te−y24texy2te−∣y∣e^{- \left|{y}\right| + \frac{\left(-1\right) \left(x - y\right)^{2}}{4 t}} = e^{- \frac{x^{2}}{4 t}} e^{- \frac{y^{2}}{4 t}} e^{\frac{x y}{2 t}} e^{- \left|{y}\right|}

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

    ∫e−x24te−y24texy2te−∣y∣ dy=e−x24t∫e−y24texy2te−∣y∣ dy\int e^{- \frac{x^{2}}{4 t}} e^{- \frac{y^{2}}{4 t}} e^{\frac{x y}{2 t}} e^{- \left|{y}\right|}\, dy = e^{- \frac{x^{2}}{4 t}} \int e^{- \frac{y^{2}}{4 t}} e^{\frac{x y}{2 t}} e^{- \left|{y}\right|}\, dy

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

      Pero la integral

      ∫e−y24texy2te−∣y∣ dy\int e^{- \frac{y^{2}}{4 t}} e^{\frac{x y}{2 t}} e^{- \left|{y}\right|}\, dy

    Por lo tanto, el resultado es: e−x24t∫e−y24texy2te−∣y∣ dye^{- \frac{x^{2}}{4 t}} \int e^{- \frac{y^{2}}{4 t}} e^{\frac{x y}{2 t}} e^{- \left|{y}\right|}\, dy

  3. Añadimos la constante de integración:

    e−x24t∫e−y24texy2te−∣y∣ dy+constante^{- \frac{x^{2}}{4 t}} \int e^{- \frac{y^{2}}{4 t}} e^{\frac{x y}{2 t}} e^{- \left|{y}\right|}\, dy+ \mathrm{constant}

Respuesta:

e−x24t∫e−y24texy2te−∣y∣ dy+constante^{- \frac{x^{2}}{4 t}} \int e^{- \frac{y^{2}}{4 t}} e^{\frac{x y}{2 t}} e^{- \left|{y}\right|}\, dy+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                           /  /                   \      
 |                            | |                    |      
 |          2                 | |          2         |    2 
 |  -(x - y)                  | |        -y    x*y   |  -x  
 |  ---------- - |y|          | |        ----  ---   |  ----
 |     4*t                    | |  -|y|  4*t   2*t   |  4*t 
 | E                 dy = C + | | e    *e    *e    dy|*e    
 |                            | |                    |      
/                             \/                     /
∫e−∣y∣+(−1)(x−y)24t dy=C+e−x24t∫e−y24texy2te−∣y∣ dy\int e^{- \left|{y}\right| + \frac{\left(-1\right) \left(x - y\right)^{2}}{4 t}}\, dy = C + e^{- \frac{x^{2}}{4 t}} \int e^{- \frac{y^{2}}{4 t}} e^{\frac{x y}{2 t}} e^{- \left|{y}\right|}\, dy
Simplificar
Respuesta [src] 
/                                                                                                     /                           2                                                 2\                                                                                                                                                                                                                                                                                                                     
|                                                                                                     |                    /    x\                                           /    x\ |                                                                                                                                                                                                                                                                                                                     
|                                                                                                     |                  t*|2 + -|                                         t*|2 + -| |                                                                                                                                                                                                                                                                                                                     
|                                                                                                     |                    \    t/                        /  ___ /    x\\    \    t/ |                                                                                                                                                                                                                                                                                                                     
|/                               2                                                       2\    2      |                  ----------                       |\/ t *|2 + -||  ----------|    2                                                                                                                                                                                                                                                                                                                
||                      /     x \                                               /     x \ |  -x       |     ___ /    x\      4             ___ /    x\    |      \    t/|      4     |  -x                                                                                                                                                                                                                                                                                                                 
||                    t*|1 - ---|                                             t*|1 - ---| |  ----     |pi*\/ t *|2 + -|*e             pi*\/ t *|2 + -|*erf|-------------|*e          |  ----                                                                                                                                                                                                                                                                                                               
||     ___ /     x \    \    2*t/         ___ /     x \    /  ___ /     x \\    \    2*t/ |  4*t      |         \    t/                        \    t/    \      2      /            |  4*t                                                                                                                                                                                                                                                                                                                
||pi*\/ t *|1 - ---|*e             - pi*\/ t *|1 - ---|*erf|\/ t *|1 - ---||*e            |*e       2*|---------------------------- - -----------------------------------------------|*e                                                                                                                                                                                                                                                                                                                   
|\         \    2*t/                          \    2*t/    \      \    2*t//              /           \             2                                        2                       /               /  /   /            pi    |   /     x \|     \     /  |   /     x \|                   pi\     /           pi    |   /     x \|     \             pi\    /   /            pi    |   /    x\|     \     /  |   /    x\|                   pi\     /           pi    |   /    x\|     \             pi\\
|------------------------------------------------------------------------------------------------ + ----------------------------------------------------------------------------------------  for And|Or|And||arg(t)| <= --, 2*|arg|1 - ---|| < pi|, And|2*|arg|1 - ---|| <= pi, |arg(t)| < --|, And||arg(t)| < --, 2*|arg|1 - ---|| < pi|, |arg(t)| < --|, Or|And||arg(t)| <= --, 2*|arg|2 + -|| < pi|, And|2*|arg|2 + -|| <= pi, |arg(t)| < --|, And||arg(t)| < --, 2*|arg|2 + -|| < pi|, |arg(t)| < --||
|                                          ____ /     x \                                                                                  ____ /    x\                                              \  \   \            2     |   \    2*t/|     /     \  |   \    2*t/|                   2 /     \           2     |   \    2*t/|     /             2 /    \   \            2     |   \    t/|     /     \  |   \    t/|                   2 /     \           2     |   \    t/|     /             2 //
|                                        \/ pi *|1 - ---|                                                                                \/ pi *|2 + -|                                                                                                                                                                                                                                                                                                                                                    
<                                               \    2*t/                                                                                       \    t/                                                                                                                                                                                                                                                                                                                                                    
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|                                                                                   oo                                                                                                                                                                                                                                                                                                                                                                                                                     
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|                                                                                   |                 2                                                                                                                                                                                                                                                                                                                                                                                                    
|                                                                                   |          (x - y)                                                                                                                                                                                                                                                                                                                                                                                                     
|                                                                                   |   -|y| - --------                                                                                                                                                                                                                                                                                                                                                                                                    
|                                                                                   |            4*t                                                                                                                                                                                                                                                                                                                                                                                                       
|                                                                                   |  e                dy                                                                                                                                                                                                                                      otherwise                                                                                                                                                  
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|                                                                                  -oo                                                                                                                                                                                                                                                                                                                                                                                                                     
\
{2(−πt(2+xt)et(2+xt)24erf⁡(t(2+xt)2)2+πt(2+xt)et(2+xt)242)e−x24tπ(2+xt)+(−πt(1−x2t)et(1−x2t)2erf⁡(t(1−x2t))+πt(1−x2t)et(1−x2t)2)e−x24tπ(1−x2t)for ((∣arg⁡(t)∣≤π2∧2∣arg⁡(1−x2t)∣<π)∨(2∣arg⁡(1−x2t)∣≤π∧∣arg⁡(t)∣<π2)∨(∣arg⁡(t)∣<π2∧2∣arg⁡(1−x2t)∣<π)∨∣arg⁡(t)∣<π2)∧((∣arg⁡(t)∣≤π2∧2∣arg⁡(2+xt)∣<π)∨(2∣arg⁡(2+xt)∣≤π∧∣arg⁡(t)∣<π2)∨(∣arg⁡(t)∣<π2∧2∣arg⁡(2+xt)∣<π)∨∣arg⁡(t)∣<π2)∫−∞∞e−∣y∣−(x−y)24t dyotherwise\begin{cases} \frac{2 \left(- \frac{\pi \sqrt{t} \left(2 + \frac{x}{t}\right) e^{\frac{t \left(2 + \frac{x}{t}\right)^{2}}{4}} \operatorname{erf}{\left(\frac{\sqrt{t} \left(2 + \frac{x}{t}\right)}{2} \right)}}{2} + \frac{\pi \sqrt{t} \left(2 + \frac{x}{t}\right) e^{\frac{t \left(2 + \frac{x}{t}\right)^{2}}{4}}}{2}\right) e^{- \frac{x^{2}}{4 t}}}{\sqrt{\pi} \left(2 + \frac{x}{t}\right)} + \frac{\left(- \pi \sqrt{t} \left(1 - \frac{x}{2 t}\right) e^{t \left(1 - \frac{x}{2 t}\right)^{2}} \operatorname{erf}{\left(\sqrt{t} \left(1 - \frac{x}{2 t}\right) \right)} + \pi \sqrt{t} \left(1 - \frac{x}{2 t}\right) e^{t \left(1 - \frac{x}{2 t}\right)^{2}}\right) e^{- \frac{x^{2}}{4 t}}}{\sqrt{\pi} \left(1 - \frac{x}{2 t}\right)} & \text{for}\: \left(\left(\left|{\arg{\left(t \right)}}\right| \leq \frac{\pi}{2} \wedge 2 \left|{\arg{\left(1 - \frac{x}{2 t} \right)}}\right| < \pi\right) \vee \left(2 \left|{\arg{\left(1 - \frac{x}{2 t} \right)}}\right| \leq \pi \wedge \left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2} \wedge 2 \left|{\arg{\left(1 - \frac{x}{2 t} \right)}}\right| < \pi\right) \vee \left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2}\right) \wedge \left(\left(\left|{\arg{\left(t \right)}}\right| \leq \frac{\pi}{2} \wedge 2 \left|{\arg{\left(2 + \frac{x}{t} \right)}}\right| < \pi\right) \vee \left(2 \left|{\arg{\left(2 + \frac{x}{t} \right)}}\right| \leq \pi \wedge \left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2} \wedge 2 \left|{\arg{\left(2 + \frac{x}{t} \right)}}\right| < \pi\right) \vee \left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2}\right) \\\int\limits_{-\infty}^{\infty} e^{- \left|{y}\right| - \frac{\left(x - y\right)^{2}}{4 t}}\, dy & \text{otherwise} \end{cases} 
=
= 
/                                                                                                     /                           2                                                 2\                                                                                                                                                                                                                                                                                                                     
|                                                                                                     |                    /    x\                                           /    x\ |                                                                                                                                                                                                                                                                                                                     
|                                                                                                     |                  t*|2 + -|                                         t*|2 + -| |                                                                                                                                                                                                                                                                                                                     
|                                                                                                     |                    \    t/                        /  ___ /    x\\    \    t/ |                                                                                                                                                                                                                                                                                                                     
|/                               2                                                       2\    2      |                  ----------                       |\/ t *|2 + -||  ----------|    2                                                                                                                                                                                                                                                                                                                
||                      /     x \                                               /     x \ |  -x       |     ___ /    x\      4             ___ /    x\    |      \    t/|      4     |  -x                                                                                                                                                                                                                                                                                                                 
||                    t*|1 - ---|                                             t*|1 - ---| |  ----     |pi*\/ t *|2 + -|*e             pi*\/ t *|2 + -|*erf|-------------|*e          |  ----                                                                                                                                                                                                                                                                                                               
||     ___ /     x \    \    2*t/         ___ /     x \    /  ___ /     x \\    \    2*t/ |  4*t      |         \    t/                        \    t/    \      2      /            |  4*t                                                                                                                                                                                                                                                                                                                
||pi*\/ t *|1 - ---|*e             - pi*\/ t *|1 - ---|*erf|\/ t *|1 - ---||*e            |*e       2*|---------------------------- - -----------------------------------------------|*e                                                                                                                                                                                                                                                                                                                   
|\         \    2*t/                          \    2*t/    \      \    2*t//              /           \             2                                        2                       /               /  /   /            pi    |   /     x \|     \     /  |   /     x \|                   pi\     /           pi    |   /     x \|     \             pi\    /   /            pi    |   /    x\|     \     /  |   /    x\|                   pi\     /           pi    |   /    x\|     \             pi\\
|------------------------------------------------------------------------------------------------ + ----------------------------------------------------------------------------------------  for And|Or|And||arg(t)| <= --, 2*|arg|1 - ---|| < pi|, And|2*|arg|1 - ---|| <= pi, |arg(t)| < --|, And||arg(t)| < --, 2*|arg|1 - ---|| < pi|, |arg(t)| < --|, Or|And||arg(t)| <= --, 2*|arg|2 + -|| < pi|, And|2*|arg|2 + -|| <= pi, |arg(t)| < --|, And||arg(t)| < --, 2*|arg|2 + -|| < pi|, |arg(t)| < --||
|                                          ____ /     x \                                                                                  ____ /    x\                                              \  \   \            2     |   \    2*t/|     /     \  |   \    2*t/|                   2 /     \           2     |   \    2*t/|     /             2 /    \   \            2     |   \    t/|     /     \  |   \    t/|                   2 /     \           2     |   \    t/|     /             2 //
|                                        \/ pi *|1 - ---|                                                                                \/ pi *|2 + -|                                                                                                                                                                                                                                                                                                                                                    
<                                               \    2*t/                                                                                       \    t/                                                                                                                                                                                                                                                                                                                                                    
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|                                                                                   oo                                                                                                                                                                                                                                                                                                                                                                                                                     
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|                                                                                   |                 2                                                                                                                                                                                                                                                                                                                                                                                                    
|                                                                                   |          (x - y)                                                                                                                                                                                                                                                                                                                                                                                                     
|                                                                                   |   -|y| - --------                                                                                                                                                                                                                                                                                                                                                                                                    
|                                                                                   |            4*t                                                                                                                                                                                                                                                                                                                                                                                                       
|                                                                                   |  e                dy                                                                                                                                                                                                                                      otherwise                                                                                                                                                  
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|                                                                                  /                                                                                                                                                                                                                                                                                                                                                                                                                       
|                                                                                  -oo                                                                                                                                                                                                                                                                                                                                                                                                                     
\
{2(−πt(2+xt)et(2+xt)24erf⁡(t(2+xt)2)2+πt(2+xt)et(2+xt)242)e−x24tπ(2+xt)+(−πt(1−x2t)et(1−x2t)2erf⁡(t(1−x2t))+πt(1−x2t)et(1−x2t)2)e−x24tπ(1−x2t)for ((∣arg⁡(t)∣≤π2∧2∣arg⁡(1−x2t)∣<π)∨(2∣arg⁡(1−x2t)∣≤π∧∣arg⁡(t)∣<π2)∨(∣arg⁡(t)∣<π2∧2∣arg⁡(1−x2t)∣<π)∨∣arg⁡(t)∣<π2)∧((∣arg⁡(t)∣≤π2∧2∣arg⁡(2+xt)∣<π)∨(2∣arg⁡(2+xt)∣≤π∧∣arg⁡(t)∣<π2)∨(∣arg⁡(t)∣<π2∧2∣arg⁡(2+xt)∣<π)∨∣arg⁡(t)∣<π2)∫−∞∞e−∣y∣−(x−y)24t dyotherwise\begin{cases} \frac{2 \left(- \frac{\pi \sqrt{t} \left(2 + \frac{x}{t}\right) e^{\frac{t \left(2 + \frac{x}{t}\right)^{2}}{4}} \operatorname{erf}{\left(\frac{\sqrt{t} \left(2 + \frac{x}{t}\right)}{2} \right)}}{2} + \frac{\pi \sqrt{t} \left(2 + \frac{x}{t}\right) e^{\frac{t \left(2 + \frac{x}{t}\right)^{2}}{4}}}{2}\right) e^{- \frac{x^{2}}{4 t}}}{\sqrt{\pi} \left(2 + \frac{x}{t}\right)} + \frac{\left(- \pi \sqrt{t} \left(1 - \frac{x}{2 t}\right) e^{t \left(1 - \frac{x}{2 t}\right)^{2}} \operatorname{erf}{\left(\sqrt{t} \left(1 - \frac{x}{2 t}\right) \right)} + \pi \sqrt{t} \left(1 - \frac{x}{2 t}\right) e^{t \left(1 - \frac{x}{2 t}\right)^{2}}\right) e^{- \frac{x^{2}}{4 t}}}{\sqrt{\pi} \left(1 - \frac{x}{2 t}\right)} & \text{for}\: \left(\left(\left|{\arg{\left(t \right)}}\right| \leq \frac{\pi}{2} \wedge 2 \left|{\arg{\left(1 - \frac{x}{2 t} \right)}}\right| < \pi\right) \vee \left(2 \left|{\arg{\left(1 - \frac{x}{2 t} \right)}}\right| \leq \pi \wedge \left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2} \wedge 2 \left|{\arg{\left(1 - \frac{x}{2 t} \right)}}\right| < \pi\right) \vee \left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2}\right) \wedge \left(\left(\left|{\arg{\left(t \right)}}\right| \leq \frac{\pi}{2} \wedge 2 \left|{\arg{\left(2 + \frac{x}{t} \right)}}\right| < \pi\right) \vee \left(2 \left|{\arg{\left(2 + \frac{x}{t} \right)}}\right| \leq \pi \wedge \left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2} \wedge 2 \left|{\arg{\left(2 + \frac{x}{t} \right)}}\right| < \pi\right) \vee \left|{\arg{\left(t \right)}}\right| < \frac{\pi}{2}\right) \\\int\limits_{-\infty}^{\infty} e^{- \left|{y}\right| - \frac{\left(x - y\right)^{2}}{4 t}}\, dy & \text{otherwise} \end{cases} 
Piecewise(((pi*sqrt(t)*(1 - x/(2*t))*exp(t*(1 - x/(2*t))^2) - pi*sqrt(t)*(1 - x/(2*t))*erf(sqrt(t)*(1 - x/(2*t)))*exp(t*(1 - x/(2*t))^2))*exp(-x^2/(4*t))/(sqrt(pi)*(1 - x/(2*t))) + 2*(pi*sqrt(t)*(2 + x/t)*exp(t*(2 + x/t)^2/4)/2 - pi*sqrt(t)*(2 + x/t)*erf(sqrt(t)*(2 + x/t)/2)*exp(t*(2 + x/t)^2/4)/2)*exp(-x^2/(4*t))/(sqrt(pi)*(2 + x/t)), ((Abs(arg(t)) < pi/2)∨((Abs(arg(t)) <= pi/2)∧(2*Abs(arg(2 + x/t)) < pi))∨((Abs(arg(t)) < pi/2)∧(2*Abs(arg(2 + x/t)) <= pi))∨((Abs(arg(t)) < pi/2)∧(2*Abs(arg(2 + x/t)) < pi)))∧((Abs(arg(t)) < pi/2)∨((Abs(arg(t)) <= pi/2)∧(2*Abs(arg(1 - x/(2*t))) < pi))∨((Abs(arg(t)) < pi/2)∧(2*Abs(arg(1 - x/(2*t))) <= pi))∨((Abs(arg(t)) < pi/2)∧(2*Abs(arg(1 - x/(2*t))) < pi)))), (Integral(exp(-|y| - (x - y)^2/(4*t)), (y, -oo, oo)), True))

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

    © Sr Examen – Калькулятор