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Resolución de integrales/ exp(a*x)/ exp(a*x)*exp(-p*x)

Integral de exp(a*x)*exp(-p*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo              
  /              
 |               
 |   a*x  -p*x   
 |  e   *e     dx
 |               
/                
0
∫0∞eaxe−px dx\int\limits_{0}^{\infty} e^{a x} e^{- p x}\, dx 
Integral(exp(a*x)*exp((-p)*x), (x, 0, oo))
Respuesta (Indefinida) [src] 
  /                    //       a*x                 \
 |                     ||     -e                    |
 |  a*x  -p*x          ||---------------  for a != p|
 | e   *e     dx = C + |<   p*x      p*x            |
 |                     ||p*e    - a*e               |
/                      ||                           |
                       \\       x         otherwise /
∫eaxe−px dx=C+{−eax−aepx+pepxfor a≠pxotherwise\int e^{a x} e^{- p x}\, dx = C + \begin{cases} - \frac{e^{a x}}{- a e^{p x} + p e^{p x}} & \text{for}\: a \neq p \\x & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/       -1                /   /            pi                  pi\     /                 pi             pi\     /                pi             pi\\
|    ---------      for Or|And||arg(p)| <= --, |pi + arg(a)| < --|, And||pi + arg(a)| <= --, |arg(p)| < --|, And||pi + arg(a)| < --, |arg(p)| < --||
|      /    p\            \   \            2                   2 /     \                 2              2 /     \                2              2 //
|    a*|1 - -|                                                                                                                                      
|      \    a/                                                                                                                                      
|                                                                                                                                                   
| oo                                                                                                                                                
<  /                                                                                                                                                
| |                                                                                                                                                 
| |   a*x  -p*x                                                                                                                                     
| |  e   *e     dx                                                             otherwise                                                            
| |                                                                                                                                                 
|/                                                                                                                                                  
|0                                                                                                                                                  
\
{−1a(1−pa)for (∣arg⁡(p)∣≤π2∧∣arg⁡(a)+π∣<π2)∨(∣arg⁡(a)+π∣≤π2∧∣arg⁡(p)∣<π2)∨(∣arg⁡(a)+π∣<π2∧∣arg⁡(p)∣<π2)∫0∞eaxe−px dxotherwise\begin{cases} - \frac{1}{a \left(1 - \frac{p}{a}\right)} & \text{for}\: \left(\left|{\arg{\left(p \right)}}\right| \leq \frac{\pi}{2} \wedge \left|{\arg{\left(a \right)} + \pi}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(a \right)} + \pi}\right| \leq \frac{\pi}{2} \wedge \left|{\arg{\left(p \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(a \right)} + \pi}\right| < \frac{\pi}{2} \wedge \left|{\arg{\left(p \right)}}\right| < \frac{\pi}{2}\right) \\\int\limits_{0}^{\infty} e^{a x} e^{- p x}\, dx & \text{otherwise} \end{cases} 
=
= 
/       -1                /   /            pi                  pi\     /                 pi             pi\     /                pi             pi\\
|    ---------      for Or|And||arg(p)| <= --, |pi + arg(a)| < --|, And||pi + arg(a)| <= --, |arg(p)| < --|, And||pi + arg(a)| < --, |arg(p)| < --||
|      /    p\            \   \            2                   2 /     \                 2              2 /     \                2              2 //
|    a*|1 - -|                                                                                                                                      
|      \    a/                                                                                                                                      
|                                                                                                                                                   
| oo                                                                                                                                                
<  /                                                                                                                                                
| |                                                                                                                                                 
| |   a*x  -p*x                                                                                                                                     
| |  e   *e     dx                                                             otherwise                                                            
| |                                                                                                                                                 
|/                                                                                                                                                  
|0                                                                                                                                                  
\
{−1a(1−pa)for (∣arg⁡(p)∣≤π2∧∣arg⁡(a)+π∣<π2)∨(∣arg⁡(a)+π∣≤π2∧∣arg⁡(p)∣<π2)∨(∣arg⁡(a)+π∣<π2∧∣arg⁡(p)∣<π2)∫0∞eaxe−px dxotherwise\begin{cases} - \frac{1}{a \left(1 - \frac{p}{a}\right)} & \text{for}\: \left(\left|{\arg{\left(p \right)}}\right| \leq \frac{\pi}{2} \wedge \left|{\arg{\left(a \right)} + \pi}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(a \right)} + \pi}\right| \leq \frac{\pi}{2} \wedge \left|{\arg{\left(p \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(a \right)} + \pi}\right| < \frac{\pi}{2} \wedge \left|{\arg{\left(p \right)}}\right| < \frac{\pi}{2}\right) \\\int\limits_{0}^{\infty} e^{a x} e^{- p x}\, dx & \text{otherwise} \end{cases} 
Piecewise((-1/(a*(1 - p/a)), ((Abs(arg(p)) <= pi/2)∧(Abs(pi + arg(a)) < pi/2))∨((Abs(arg(p)) < pi/2)∧(Abs(pi + arg(a)) <= pi/2))∨((Abs(arg(p)) < pi/2)∧(Abs(pi + arg(a)) < pi/2))), (Integral(exp(a*x)*exp(-p*x), (x, 0, oo)), True))

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

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