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Resolución de integrales/ cos(w*t)/ (1-w/a)*cos(w*t)

Integral de (1-w/a)*cos(w*t) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                    
  /                    
 |                     
 |  /    w\            
 |  |1 - -|*cos(w*t) dw
 |  \    a/            
 |                     
/                      
0
$$\int\limits_{0}^{1} \left(1 - \frac{w}{a}\right) \cos{\left(t w \right)}\, dw$$ 
Integral((1 - w/a)*cos(w*t), (w, 0, 1))
Respuesta (Indefinida) [src] 
                               //           2                      \                                                      
                               ||          w                       |                                                      
                               ||          --             for t = 0|                                                      
                               ||          2                       |                                                      
                               ||                                  |     //   w      for t = 0\     //   w      for t = 0\
                               ||/-cos(t*w)                        |     ||                   |     ||                   |
                             - |<|----------  for t != 0           | + w*|
$$\int \left(1 - \frac{w}{a}\right) \cos{\left(t w \right)}\, dw = C - \frac{- a \left(\begin{cases} w & \text{for}\: t = 0 \\\frac{\sin{\left(t w \right)}}{t} & \text{otherwise} \end{cases}\right) + w \left(\begin{cases} w & \text{for}\: t = 0 \\\frac{\sin{\left(t w \right)}}{t} & \text{otherwise} \end{cases}\right) - \begin{cases} \frac{w^{2}}{2} & \text{for}\: t = 0 \\\frac{\begin{cases} - \frac{\cos{\left(t w \right)}}{t} & \text{for}\: t \neq 0 \\0 & \text{otherwise} \end{cases}}{t} & \text{otherwise} \end{cases}}{a}$$
Simplificar
Respuesta [src] 
/ 1     sin(t)   sin(t)   cos(t)                                  
|---- + ------ - ------ - ------  for And(t > -oo, t < oo, t != 0)
|   2     t       a*t         2                                   
|a*t                       a*t                                    
<                                                                 
|                 1                                               
|            1 - ---                         otherwise            
|                2*a                                              
\
$$\begin{cases} \frac{\sin{\left(t \right)}}{t} - \frac{\sin{\left(t \right)}}{a t} - \frac{\cos{\left(t \right)}}{a t^{2}} + \frac{1}{a t^{2}} & \text{for}\: t > -\infty \wedge t < \infty \wedge t \neq 0 \\1 - \frac{1}{2 a} & \text{otherwise} \end{cases}$$ 
=
= 
/ 1     sin(t)   sin(t)   cos(t)                                  
|---- + ------ - ------ - ------  for And(t > -oo, t < oo, t != 0)
|   2     t       a*t         2                                   
|a*t                       a*t                                    
<                                                                 
|                 1                                               
|            1 - ---                         otherwise            
|                2*a                                              
\
$$\begin{cases} \frac{\sin{\left(t \right)}}{t} - \frac{\sin{\left(t \right)}}{a t} - \frac{\cos{\left(t \right)}}{a t^{2}} + \frac{1}{a t^{2}} & \text{for}\: t > -\infty \wedge t < \infty \wedge t \neq 0 \\1 - \frac{1}{2 a} & \text{otherwise} \end{cases}$$ 
Piecewise((1/(a*t^2) + sin(t)/t - sin(t)/(a*t) - cos(t)/(a*t^2), (t > -oo)∧(t < oo)∧(Ne(t, 0))), (1 - 1/(2*a), True))

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

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