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Resolución de integrales/ cosbx/ e^(-ax)/ e^(-ax)*cosbxdx

Integral de e^(-ax)*cosbxdx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo                  
  /                  
 |                   
 |   -a*x            
 |  E    *cos(b*x) dx
 |                   
/                    
0
$$\int\limits_{0}^{\infty} e^{- a x} \cos{\left(b x \right)}\, dx$$ 
Integral(E^((-a)*x)*cos(b*x), (x, 0, oo))
Respuesta (Indefinida) [src] 
                           //                             x                                for And(a = 0, b = 0)\
                           ||                                                                                   |
                           ||             I*b*x    I*b*x                 I*b*x                                  |
                           || x*cos(b*x)*e        e     *sin(b*x)   I*x*e     *sin(b*x)                         |
                           || ----------------- + --------------- - -------------------        for a = -I*b     |
  /                        ||         2                 2*b                  2                                  |
 |                         ||                                                                                   |
 |  -a*x                   ||            -I*b*x    -I*b*x                 -I*b*x                                |
 | E    *cos(b*x) dx = C + |
$$\int e^{- a x} \cos{\left(b x \right)}\, dx = C + \begin{cases} x & \text{for}\: a = 0 \wedge b = 0 \\- \frac{i x e^{i b x} \sin{\left(b x \right)}}{2} + \frac{x e^{i b x} \cos{\left(b x \right)}}{2} + \frac{e^{i b x} \sin{\left(b x \right)}}{2 b} & \text{for}\: a = - i b \\\frac{i x e^{- i b x} \sin{\left(b x \right)}}{2} + \frac{x e^{- i b x} \cos{\left(b x \right)}}{2} + \frac{e^{- i b x} \sin{\left(b x \right)}}{2 b} & \text{for}\: a = i b \\- \frac{a \cos{\left(b x \right)}}{a^{2} e^{a x} + b^{2} e^{a x}} + \frac{b \sin{\left(b x \right)}}{a^{2} e^{a x} + b^{2} e^{a x}} & \text{otherwise} \end{cases}$$
Simplificar
Respuesta [src] 
/         1                                                     
|     ----------        for And(2*|arg(b)| = 0, 2*|arg(a)| < pi)
|       /     2\                                                
|       |    b |                                                
|     a*|1 + --|                                                
|       |     2|                                                
|       \    a /                                                
|                                                               
< oo                                                            
|  /                                                            
| |                                                             
| |            -a*x                                             
| |  cos(b*x)*e     dx                 otherwise                
| |                                                             
|/                                                              
|0                                                              
\
$$\begin{cases} \frac{1}{a \left(1 + \frac{b^{2}}{a^{2}}\right)} & \text{for}\: 2 \left|{\arg{\left(b \right)}}\right| = 0 \wedge 2 \left|{\arg{\left(a \right)}}\right| < \pi \\\int\limits_{0}^{\infty} e^{- a x} \cos{\left(b x \right)}\, dx & \text{otherwise} \end{cases}$$ 
=
= 
/         1                                                     
|     ----------        for And(2*|arg(b)| = 0, 2*|arg(a)| < pi)
|       /     2\                                                
|       |    b |                                                
|     a*|1 + --|                                                
|       |     2|                                                
|       \    a /                                                
|                                                               
< oo                                                            
|  /                                                            
| |                                                             
| |            -a*x                                             
| |  cos(b*x)*e     dx                 otherwise                
| |                                                             
|/                                                              
|0                                                              
\
$$\begin{cases} \frac{1}{a \left(1 + \frac{b^{2}}{a^{2}}\right)} & \text{for}\: 2 \left|{\arg{\left(b \right)}}\right| = 0 \wedge 2 \left|{\arg{\left(a \right)}}\right| < \pi \\\int\limits_{0}^{\infty} e^{- a x} \cos{\left(b x \right)}\, dx & \text{otherwise} \end{cases}$$ 
Piecewise((1/(a*(1 + b^2/a^2)), (2*Abs(arg(b)) = 0))∧(2*Abs(arg(a)) < pi), (Integral(cos(b*x)*exp(-a*x), (x, 0, oo)), True))

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

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