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Resolución de integrales/ e^(a*x)/ cos(w*x)/ e^(a*x)*cos(w*x)

Integral de e^(a*x)*cos(w*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  0                 
  /                 
 |                  
 |   a*x            
 |  E   *cos(w*x) dx
 |                  
/                   
-oo
∫−∞0eaxcos⁡(wx) dx\int\limits_{-\infty}^{0} e^{a x} \cos{\left(w x \right)}\, dx 
Integral(E^(a*x)*cos(w*x), (x, -oo, 0))
Respuesta (Indefinida) [src] 
                          //                              x                                 for And(a = 0, w = 0)\
                          ||                                                                                     |
                          ||            -I*w*x        -I*w*x                        -I*w*x                       |
                          ||x*cos(w*x)*e         I*x*e      *sin(w*x)   I*cos(w*x)*e                             |
                          ||------------------ + -------------------- + ------------------      for a = -I*w     |
  /                       ||        2                     2                    2*w                               |
 |                        ||                                                                                     |
 |  a*x                   ||             I*w*x        I*w*x                        I*w*x                         |
 | E   *cos(w*x) dx = C + |< x*cos(w*x)*e        I*x*e     *sin(w*x)   I*cos(w*x)*e                              |
 |                        || ----------------- - ------------------- - -----------------         for a = I*w     |
/                         ||         2                    2                   2*w                                |
                          ||                                                                                     |
                          ||                          a*x      a*x                                               |
                          ||              a*cos(w*x)*e      w*e   *sin(w*x)                                      |
                          ||              --------------- + ---------------                       otherwise      |
                          ||                   2    2            2    2                                          |
                          \\                  a  + w            a  + w                                           /
∫eaxcos⁡(wx) dx=C+{xfor a=0∧w=0ixe−iwxsin⁡(wx)2+xe−iwxcos⁡(wx)2+ie−iwxcos⁡(wx)2wfor a=−iw−ixeiwxsin⁡(wx)2+xeiwxcos⁡(wx)2−ieiwxcos⁡(wx)2wfor a=iwaeaxcos⁡(wx)a2+w2+weaxsin⁡(wx)a2+w2otherwise\int e^{a x} \cos{\left(w x \right)}\, dx = C + \begin{cases} x & \text{for}\: a = 0 \wedge w = 0 \\\frac{i x e^{- i w x} \sin{\left(w x \right)}}{2} + \frac{x e^{- i w x} \cos{\left(w x \right)}}{2} + \frac{i e^{- i w x} \cos{\left(w x \right)}}{2 w} & \text{for}\: a = - i w \\- \frac{i x e^{i w x} \sin{\left(w x \right)}}{2} + \frac{x e^{i w x} \cos{\left(w x \right)}}{2} - \frac{i e^{i w x} \cos{\left(w x \right)}}{2 w} & \text{for}\: a = i w \\\frac{a e^{a x} \cos{\left(w x \right)}}{a^{2} + w^{2}} + \frac{w e^{a x} \sin{\left(w x \right)}}{a^{2} + w^{2}} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/         1                                                    
|     ----------       for And(2*|arg(w)| = 0, 2*|arg(a)| < pi)
|       /     2\                                               
|       |    w |                                               
|     a*|1 + --|                                               
|       |     2|                                               
|       \    a /                                               
|                                                              
<  0                                                           
|  /                                                           
| |                                                            
| |            a*x                                             
| |  cos(w*x)*e    dx                 otherwise                
| |                                                            
|/                                                             
|-oo                                                           
\
{1a(1+w2a2)for 2∣arg⁡(w)∣=0∧2∣arg⁡(a)∣<π∫−∞0eaxcos⁡(wx) dxotherwise\begin{cases} \frac{1}{a \left(1 + \frac{w^{2}}{a^{2}}\right)} & \text{for}\: 2 \left|{\arg{\left(w \right)}}\right| = 0 \wedge 2 \left|{\arg{\left(a \right)}}\right| < \pi \\\int\limits_{-\infty}^{0} e^{a x} \cos{\left(w x \right)}\, dx & \text{otherwise} \end{cases} 
=
= 
/         1                                                    
|     ----------       for And(2*|arg(w)| = 0, 2*|arg(a)| < pi)
|       /     2\                                               
|       |    w |                                               
|     a*|1 + --|                                               
|       |     2|                                               
|       \    a /                                               
|                                                              
<  0                                                           
|  /                                                           
| |                                                            
| |            a*x                                             
| |  cos(w*x)*e    dx                 otherwise                
| |                                                            
|/                                                             
|-oo                                                           
\
{1a(1+w2a2)for 2∣arg⁡(w)∣=0∧2∣arg⁡(a)∣<π∫−∞0eaxcos⁡(wx) dxotherwise\begin{cases} \frac{1}{a \left(1 + \frac{w^{2}}{a^{2}}\right)} & \text{for}\: 2 \left|{\arg{\left(w \right)}}\right| = 0 \wedge 2 \left|{\arg{\left(a \right)}}\right| < \pi \\\int\limits_{-\infty}^{0} e^{a x} \cos{\left(w x \right)}\, dx & \text{otherwise} \end{cases} 
Piecewise((1/(a*(1 + w^2/a^2)), (2*Abs(arg(w)) = 0))∧(2*Abs(arg(a)) < pi), (Integral(cos(w*x)*exp(a*x), (x, -oo, 0)), True))

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

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