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Resolución de integrales/ exp(-a*x)/ exp(-a*x)*cos(f*x)

Integral de exp(-a*x)*cos(f*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                  
  /                  
 |                   
 |   -a*x            
 |  e    *cos(f*x) dx
 |                   
/                    
0
∫01e−axcos⁡(fx) dx\int\limits_{0}^{1} e^{- a x} \cos{\left(f x \right)}\, dx 
Integral(exp((-a)*x)*cos(f*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
                             //                           /sin(f*x)   x*cos(f*x)                                                  \                                 
                             ||                           |-------- - ----------  for f != 0                                      |                                 
                             ||                           |    2          f                                                       |                                 
                             ||                           <   f                                                          for a = 0|                                 
                             ||                           |                                                                       |                                 
                             ||                           |          0            otherwise                                       |                                 
                             ||                           \                                                                       |                                 
                             ||                                                                                                   |                                 
                             || //                             0                                for And(a = 0, f = 0)\            |                                 
                             || ||                                                                                   |            |                                 
  /                          || ||    I*f*x                      I*f*x                 I*f*x                         |            |   //   x     for a = 0\         
 |                           || || x*e     *sin(f*x)   cos(f*x)*e        I*x*cos(f*x)*e                              |            |   ||                  |         
 |  -a*x                     || || ----------------- - --------------- + -------------------        for a = -I*f     |            |   ||  -a*x            |         
 | e    *cos(f*x) dx = C + f*|< ||         2                 2*f                  2                                  |            | + |<-e                |*cos(f*x)
 |                           || ||                                                                                   |            |   ||-------  otherwise|         
/                            || ||   -I*f*x                      -I*f*x                 -I*f*x                       |            |   ||   a              |         
                             ||-|
∫e−axcos⁡(fx) dx=C+f({{−xcos⁡(fx)f+sin⁡(fx)f2for f≠00otherwisefor a=0−{0for a=0∧f=0xeifxsin⁡(fx)2+ixeifxcos⁡(fx)2−eifxcos⁡(fx)2ffor a=−ifxe−ifxsin⁡(fx)2−ixe−ifxcos⁡(fx)2−e−ifxcos⁡(fx)2ffor a=if−asin⁡(fx)a2eax+f2eax−fcos⁡(fx)a2eax+f2eaxotherwiseaotherwise)+({xfor a=0−e−axaotherwise)cos⁡(fx)\int e^{- a x} \cos{\left(f x \right)}\, dx = C + f \left(\begin{cases} \begin{cases} - \frac{x \cos{\left(f x \right)}}{f} + \frac{\sin{\left(f x \right)}}{f^{2}} & \text{for}\: f \neq 0 \\0 & \text{otherwise} \end{cases} & \text{for}\: a = 0 \\- \frac{\begin{cases} 0 & \text{for}\: a = 0 \wedge f = 0 \\\frac{x e^{i f x} \sin{\left(f x \right)}}{2} + \frac{i x e^{i f x} \cos{\left(f x \right)}}{2} - \frac{e^{i f x} \cos{\left(f x \right)}}{2 f} & \text{for}\: a = - i f \\\frac{x e^{- i f x} \sin{\left(f x \right)}}{2} - \frac{i x e^{- i f x} \cos{\left(f x \right)}}{2} - \frac{e^{- i f x} \cos{\left(f x \right)}}{2 f} & \text{for}\: a = i f \\- \frac{a \sin{\left(f x \right)}}{a^{2} e^{a x} + f^{2} e^{a x}} - \frac{f \cos{\left(f x \right)}}{a^{2} e^{a x} + f^{2} e^{a x}} & \text{otherwise} \end{cases}}{a} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} x & \text{for}\: a = 0 \\- \frac{e^{- a x}}{a} & \text{otherwise} \end{cases}\right) \cos{\left(f x \right)}
Simplificar
Respuesta [src] 
/                     1                                     for Or(And(a = 0, f = 0), And(a = 0, a = -I*f, f = 0), And(a = 0, a = I*f, f = 0), And(a = 0, a = -I*f, a = I*f, f = 0))             
|                                                                                                                                                                                                
|         I*f    I*f             I*f                                                                                                                                                             
| cos(f)*e      e   *sin(f)   I*e   *sin(f)                                                                                                                                                      
| ----------- + ----------- - -------------    for Or(And(a = 0, a = -I*f), And(a = -I*f, a = I*f), And(a = -I*f, f = 0), And(a = 0, a = -I*f, a = I*f), And(a = -I*f, a = I*f, f = 0), a = -I*f)
|      2            2*f             2                                                                                                                                                            
|                                                                                                                                                                                                
|        -I*f      -I*f           -I*f
{1for (a=0∧f=0)∨(a=0∧a=−if∧f=0)∨(a=0∧a=if∧f=0)∨(a=0∧a=−if∧a=if∧f=0)−ieifsin⁡(f)2+eifcos⁡(f)2+eifsin⁡(f)2ffor (a=0∧a=−if)∨(a=−if∧a=if)∨(a=−if∧f=0)∨(a=0∧a=−if∧a=if)∨(a=−if∧a=if∧f=0)∨a=−ifie−ifsin⁡(f)2+e−ifcos⁡(f)2+e−ifsin⁡(f)2ffor (a=0∧a=if)∨(a=if∧f=0)∨a=if−acos⁡(f)a2ea+f2ea+aa2+f2+fsin⁡(f)a2ea+f2eaotherwise\begin{cases} 1 & \text{for}\: \left(a = 0 \wedge f = 0\right) \vee \left(a = 0 \wedge a = - i f \wedge f = 0\right) \vee \left(a = 0 \wedge a = i f \wedge f = 0\right) \vee \left(a = 0 \wedge a = - i f \wedge a = i f \wedge f = 0\right) \\- \frac{i e^{i f} \sin{\left(f \right)}}{2} + \frac{e^{i f} \cos{\left(f \right)}}{2} + \frac{e^{i f} \sin{\left(f \right)}}{2 f} & \text{for}\: \left(a = 0 \wedge a = - i f\right) \vee \left(a = - i f \wedge a = i f\right) \vee \left(a = - i f \wedge f = 0\right) \vee \left(a = 0 \wedge a = - i f \wedge a = i f\right) \vee \left(a = - i f \wedge a = i f \wedge f = 0\right) \vee a = - i f \\\frac{i e^{- i f} \sin{\left(f \right)}}{2} + \frac{e^{- i f} \cos{\left(f \right)}}{2} + \frac{e^{- i f} \sin{\left(f \right)}}{2 f} & \text{for}\: \left(a = 0 \wedge a = i f\right) \vee \left(a = i f \wedge f = 0\right) \vee a = i f \\- \frac{a \cos{\left(f \right)}}{a^{2} e^{a} + f^{2} e^{a}} + \frac{a}{a^{2} + f^{2}} + \frac{f \sin{\left(f \right)}}{a^{2} e^{a} + f^{2} e^{a}} & \text{otherwise} \end{cases} 
=
= 
/                     1                                     for Or(And(a = 0, f = 0), And(a = 0, a = -I*f, f = 0), And(a = 0, a = I*f, f = 0), And(a = 0, a = -I*f, a = I*f, f = 0))             
|                                                                                                                                                                                                
|         I*f    I*f             I*f                                                                                                                                                             
| cos(f)*e      e   *sin(f)   I*e   *sin(f)                                                                                                                                                      
| ----------- + ----------- - -------------    for Or(And(a = 0, a = -I*f), And(a = -I*f, a = I*f), And(a = -I*f, f = 0), And(a = 0, a = -I*f, a = I*f), And(a = -I*f, a = I*f, f = 0), a = -I*f)
|      2            2*f             2                                                                                                                                                            
|                                                                                                                                                                                                
|        -I*f      -I*f           -I*f
{1for (a=0∧f=0)∨(a=0∧a=−if∧f=0)∨(a=0∧a=if∧f=0)∨(a=0∧a=−if∧a=if∧f=0)−ieifsin⁡(f)2+eifcos⁡(f)2+eifsin⁡(f)2ffor (a=0∧a=−if)∨(a=−if∧a=if)∨(a=−if∧f=0)∨(a=0∧a=−if∧a=if)∨(a=−if∧a=if∧f=0)∨a=−ifie−ifsin⁡(f)2+e−ifcos⁡(f)2+e−ifsin⁡(f)2ffor (a=0∧a=if)∨(a=if∧f=0)∨a=if−acos⁡(f)a2ea+f2ea+aa2+f2+fsin⁡(f)a2ea+f2eaotherwise\begin{cases} 1 & \text{for}\: \left(a = 0 \wedge f = 0\right) \vee \left(a = 0 \wedge a = - i f \wedge f = 0\right) \vee \left(a = 0 \wedge a = i f \wedge f = 0\right) \vee \left(a = 0 \wedge a = - i f \wedge a = i f \wedge f = 0\right) \\- \frac{i e^{i f} \sin{\left(f \right)}}{2} + \frac{e^{i f} \cos{\left(f \right)}}{2} + \frac{e^{i f} \sin{\left(f \right)}}{2 f} & \text{for}\: \left(a = 0 \wedge a = - i f\right) \vee \left(a = - i f \wedge a = i f\right) \vee \left(a = - i f \wedge f = 0\right) \vee \left(a = 0 \wedge a = - i f \wedge a = i f\right) \vee \left(a = - i f \wedge a = i f \wedge f = 0\right) \vee a = - i f \\\frac{i e^{- i f} \sin{\left(f \right)}}{2} + \frac{e^{- i f} \cos{\left(f \right)}}{2} + \frac{e^{- i f} \sin{\left(f \right)}}{2 f} & \text{for}\: \left(a = 0 \wedge a = i f\right) \vee \left(a = i f \wedge f = 0\right) \vee a = i f \\- \frac{a \cos{\left(f \right)}}{a^{2} e^{a} + f^{2} e^{a}} + \frac{a}{a^{2} + f^{2}} + \frac{f \sin{\left(f \right)}}{a^{2} e^{a} + f^{2} e^{a}} & \text{otherwise} \end{cases} 
Piecewise((1, ((a = 0)∧(f = 0))∨((a = 0)∧(f = 0)∧(a = i*f))∨((a = 0)∧(f = 0)∧(a = -i*f))∨((a = 0)∧(f = 0)∧(a = i*f)∧(a = -i*f))), (cos(f)*exp(i*f)/2 + exp(i*f)*sin(f)/(2*f) - i*exp(i*f)*sin(f)/2, (a = -i*f)∨((a = 0)∧(a = -i*f))∨((f = 0)∧(a = -i*f))∨((a = i*f)∧(a = -i*f))∨((a = 0)∧(a = i*f)∧(a = -i*f))∨((f = 0)∧(a = i*f)∧(a = -i*f))), (cos(f)*exp(-i*f)/2 + i*exp(-i*f)*sin(f)/2 + exp(-i*f)*sin(f)/(2*f), (a = i*f)∨((a = 0)∧(a = i*f))∨((f = 0)∧(a = i*f))), (a/(a^2 + f^2) + f*sin(f)/(a^2*exp(a) + f^2*exp(a)) - a*cos(f)/(a^2*exp(a) + f^2*exp(a)), True))

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

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