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Resolución de integrales/ (t-x)/ sin(w*x)/ exp(-a*(t-x))*sin(w*x)

Integral de exp(-a*(t-x))*sin(w*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  t                        
  /                        
 |                         
 |   -a*(t - x)            
 |  e          *sin(w*x) dx
 |                         
/                          
0
$$\int\limits_{0}^{t} e^{- a \left(t - x\right)} \sin{\left(w x \right)}\, dx$$ 
Integral(exp((-a)*(t - x))*sin(w*x), (x, 0, t))
Respuesta (Indefinida) [src] 
                                 /                                //                          /cos(w*x)   x*sin(w*x)                                                 \\      
                                 |                                ||                          |-------- + ----------  for w != 0                                     ||      
                                 |                                ||                          |    2          w                                                      ||      
                                 |                                ||                          |   w                                                                  ||      
                                 |                                ||                          <                                                             for a = 0||      
                                 |                                ||                          |          2                                                           ||      
                                 |                                ||                          |         x                                                            ||      
                                 |                                ||                          |         --            otherwise                                      ||      
                                 |                                ||                          \         2                                                            ||      
                                 |                                ||                                                                                                 ||      
                                 |                                ||/                              x                                 for And(a = 0, w = 0)           ||      
  /                              |// x    for a = 0\              |||                                                                                                ||      
 |                               |||               |              |||            -I*w*x        -I*w*x                        -I*w*x                                  ||      
 |  -a*(t - x)                   ||| a*x           |              |||x*cos(w*x)*e         I*x*e      *sin(w*x)   I*cos(w*x)*e                                        ||  -a*t
 | e          *sin(w*x) dx = C + ||
$$\int e^{- a \left(t - x\right)} \sin{\left(w x \right)}\, dx = C + \left(- w \left(\begin{cases} \begin{cases} \frac{x \sin{\left(w x \right)}}{w} + \frac{\cos{\left(w x \right)}}{w^{2}} & \text{for}\: w \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases} & \text{for}\: a = 0 \\\frac{\begin{cases} x & \text{for}\: a = 0 \wedge w = 0 \\\frac{i x e^{- i w x} \sin{\left(w x \right)}}{2} + \frac{x e^{- i w x} \cos{\left(w x \right)}}{2} + \frac{i e^{- i w x} \cos{\left(w x \right)}}{2 w} & \text{for}\: a = - i w \\- \frac{i x e^{i w x} \sin{\left(w x \right)}}{2} + \frac{x e^{i w x} \cos{\left(w x \right)}}{2} - \frac{i e^{i w x} \cos{\left(w x \right)}}{2 w} & \text{for}\: a = i w \\\frac{a e^{a x} \cos{\left(w x \right)}}{a^{2} + w^{2}} + \frac{w e^{a x} \sin{\left(w x \right)}}{a^{2} + w^{2}} & \text{otherwise} \end{cases}}{a} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} x & \text{for}\: a = 0 \\\frac{e^{a x}}{a} & \text{otherwise} \end{cases}\right) \sin{\left(w x \right)}\right) e^{- a t}$$
Simplificar
Respuesta [src] 
/                      0                                      for Or(And(a = 0, w = 0), And(a = 0, a = -I*w, w = 0), And(a = 0, a = I*w, w = 0), And(a = 0, a = -I*w, a = I*w, w = 0))             
|                                                                                                                                                                                                  
|              I*t*w                                                                                                                                                                               
|t*sin(t*w)   e        cos(t*w)   I*t*cos(t*w)                                                                                                                                                     
|---------- + ------ - -------- - ------------   for Or(And(a = 0, a = -I*w), And(a = -I*w, a = I*w), And(a = -I*w, w = 0), And(a = 0, a = -I*w, a = I*w), And(a = -I*w, a = I*w, w = 0), a = -I*w)
|    2         2*w       2*w           2                                                                                                                                                           
|                                                                                                                                                                                                  
|              -I*t*w
$$\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge w = 0\right) \vee \left(a = 0 \wedge a = - i w \wedge w = 0\right) \vee \left(a = 0 \wedge a = i w \wedge w = 0\right) \vee \left(a = 0 \wedge a = - i w \wedge a = i w \wedge w = 0\right) \\\frac{t \sin{\left(t w \right)}}{2} - \frac{i t \cos{\left(t w \right)}}{2} + \frac{e^{i t w}}{2 w} - \frac{\cos{\left(t w \right)}}{2 w} & \text{for}\: \left(a = 0 \wedge a = - i w\right) \vee \left(a = - i w \wedge a = i w\right) \vee \left(a = - i w \wedge w = 0\right) \vee \left(a = 0 \wedge a = - i w \wedge a = i w\right) \vee \left(a = - i w \wedge a = i w \wedge w = 0\right) \vee a = - i w \\\frac{t \sin{\left(t w \right)}}{2} + \frac{i t \cos{\left(t w \right)}}{2} - \frac{\cos{\left(t w \right)}}{2 w} + \frac{e^{- i t w}}{2 w} & \text{for}\: \left(a = 0 \wedge a = i w\right) \vee \left(a = i w \wedge w = 0\right) \vee a = i w \\\frac{a \sin{\left(t w \right)}}{a^{2} + w^{2}} + \frac{w}{a^{2} e^{a t} + w^{2} e^{a t}} - \frac{w \cos{\left(t w \right)}}{a^{2} + w^{2}} & \text{otherwise} \end{cases}$$ 
=
= 
/                      0                                      for Or(And(a = 0, w = 0), And(a = 0, a = -I*w, w = 0), And(a = 0, a = I*w, w = 0), And(a = 0, a = -I*w, a = I*w, w = 0))             
|                                                                                                                                                                                                  
|              I*t*w                                                                                                                                                                               
|t*sin(t*w)   e        cos(t*w)   I*t*cos(t*w)                                                                                                                                                     
|---------- + ------ - -------- - ------------   for Or(And(a = 0, a = -I*w), And(a = -I*w, a = I*w), And(a = -I*w, w = 0), And(a = 0, a = -I*w, a = I*w), And(a = -I*w, a = I*w, w = 0), a = -I*w)
|    2         2*w       2*w           2                                                                                                                                                           
|                                                                                                                                                                                                  
|              -I*t*w
$$\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge w = 0\right) \vee \left(a = 0 \wedge a = - i w \wedge w = 0\right) \vee \left(a = 0 \wedge a = i w \wedge w = 0\right) \vee \left(a = 0 \wedge a = - i w \wedge a = i w \wedge w = 0\right) \\\frac{t \sin{\left(t w \right)}}{2} - \frac{i t \cos{\left(t w \right)}}{2} + \frac{e^{i t w}}{2 w} - \frac{\cos{\left(t w \right)}}{2 w} & \text{for}\: \left(a = 0 \wedge a = - i w\right) \vee \left(a = - i w \wedge a = i w\right) \vee \left(a = - i w \wedge w = 0\right) \vee \left(a = 0 \wedge a = - i w \wedge a = i w\right) \vee \left(a = - i w \wedge a = i w \wedge w = 0\right) \vee a = - i w \\\frac{t \sin{\left(t w \right)}}{2} + \frac{i t \cos{\left(t w \right)}}{2} - \frac{\cos{\left(t w \right)}}{2 w} + \frac{e^{- i t w}}{2 w} & \text{for}\: \left(a = 0 \wedge a = i w\right) \vee \left(a = i w \wedge w = 0\right) \vee a = i w \\\frac{a \sin{\left(t w \right)}}{a^{2} + w^{2}} + \frac{w}{a^{2} e^{a t} + w^{2} e^{a t}} - \frac{w \cos{\left(t w \right)}}{a^{2} + w^{2}} & \text{otherwise} \end{cases}$$ 
Piecewise((0, ((a = 0)∧(w = 0))∨((a = 0)∧(w = 0)∧(a = i*w))∨((a = 0)∧(w = 0)∧(a = -i*w))∨((a = 0)∧(w = 0)∧(a = i*w)∧(a = -i*w))), (t*sin(t*w)/2 + exp(i*t*w)/(2*w) - cos(t*w)/(2*w) - i*t*cos(t*w)/2, (a = -i*w)∨((a = 0)∧(a = -i*w))∨((w = 0)∧(a = -i*w))∨((a = i*w)∧(a = -i*w))∨((a = 0)∧(a = i*w)∧(a = -i*w))∨((w = 0)∧(a = i*w)∧(a = -i*w))), (t*sin(t*w)/2 + exp(-i*t*w)/(2*w) - cos(t*w)/(2*w) + i*t*cos(t*w)/2, (a = i*w)∨((a = 0)∧(a = i*w))∨((w = 0)∧(a = i*w))), (w/(a^2*exp(a*t) + w^2*exp(a*t)) + a*sin(t*w)/(a^2 + w^2) - w*cos(t*w)/(a^2 + w^2), True))

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

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