Sr Examen

Otras calculadoras

Resolución de integrales/ cos(w*t)/ cos(w*t)*exp(-i*w*t)

Integral de cos(w*t)*exp(-i*w*t) dt

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 t2                    
  /                    
 |                     
 |            -I*w*t   
 |  cos(w*t)*e       dt
 |                     
/                      
0
∫0t2et−iwcos⁡(tw) dt\int\limits_{0}^{t_{2}} e^{t - i w} \cos{\left(t w \right)}\, dt 
Integral(cos(w*t)*exp(((-i)*w)*t), (t, 0, t2))
Respuesta (Indefinida) [src] 
                               ///         /t*w\                                   2/t*w\                         \                                   
                               |||    2*tan|---|                            t*w*tan |---|                         |                                   
                               |||         \ 2 /             t*w                    \ 2 /                         |                                   
                               |||----------------- - ----------------- + -----------------  for w != 0           |                                   
                               ||< 2    2    2/t*w\    2    2    2/t*w\    2    2    2/t*w\              for w = 0|                                   
                               |||w  + w *tan |---|   w  + w *tan |---|   w  + w *tan |---|                       |                                   
  /                            |||            \ 2 /               \ 2 /               \ 2 /                       |   //    t      for w = 0\         
 |                             |||                                                                                |   ||                    |         
 |           -I*w*t            ||\                            0                              otherwise            |   ||   -I*t*w           |         
 | cos(w*t)*e       dt = C + w*|<                                                                                 | + |
∫et−iwcos⁡(tw) dt=C+w({{twtan⁡2(tw2)w2tan⁡2(tw2)+w2−tww2tan⁡2(tw2)+w2+2tan⁡(tw2)w2tan⁡2(tw2)+w2for w≠00otherwisefor w=0t2w+{−ie−2itw4w2for 4w2≠0−t2wotherwiseotherwise)+({tfor w=0ie−itwwotherwise)cos⁡(tw)\int e^{t - i w} \cos{\left(t w \right)}\, dt = C + w \left(\begin{cases} \begin{cases} \frac{t w \tan^{2}{\left(\frac{t w}{2} \right)}}{w^{2} \tan^{2}{\left(\frac{t w}{2} \right)} + w^{2}} - \frac{t w}{w^{2} \tan^{2}{\left(\frac{t w}{2} \right)} + w^{2}} + \frac{2 \tan{\left(\frac{t w}{2} \right)}}{w^{2} \tan^{2}{\left(\frac{t w}{2} \right)} + w^{2}} & \text{for}\: w \neq 0 \\0 & \text{otherwise} \end{cases} & \text{for}\: w = 0 \\\frac{t}{2 w} + \begin{cases} - \frac{i e^{- 2 i t w}}{4 w^{2}} & \text{for}\: 4 w^{2} \neq 0 \\- \frac{t}{2 w} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} t & \text{for}\: w = 0 \\\frac{i e^{- i t w}}{w} & \text{otherwise} \end{cases}\right) \cos{\left(t w \right)}
Simplificar
Respuesta [src] 
     //           -2*I*t2*w                                  \
     ||   I    I*e                                           |
     ||- --- + ------------  for And(w > -oo, w < oo, w != 0)|
t2   ||  4*w       4*w                                       |
-- + |<                                                      |
2    ||         t2                                           |
     ||         --                      otherwise            |
     ||         2                                            |
     \\                                                      /
t22+{−i4w+ie−2it2w4wfor w>−∞∧w<∞∧w≠0t22otherwise\frac{t_{2}}{2} + \begin{cases} - \frac{i}{4 w} + \frac{i e^{- 2 i t_{2} w}}{4 w} & \text{for}\: w > -\infty \wedge w < \infty \wedge w \neq 0 \\\frac{t_{2}}{2} & \text{otherwise} \end{cases} 
=
= 
     //           -2*I*t2*w                                  \
     ||   I    I*e                                           |
     ||- --- + ------------  for And(w > -oo, w < oo, w != 0)|
t2   ||  4*w       4*w                                       |
-- + |<                                                      |
2    ||         t2                                           |
     ||         --                      otherwise            |
     ||         2                                            |
     \\                                                      /
t22+{−i4w+ie−2it2w4wfor w>−∞∧w<∞∧w≠0t22otherwise\frac{t_{2}}{2} + \begin{cases} - \frac{i}{4 w} + \frac{i e^{- 2 i t_{2} w}}{4 w} & \text{for}\: w > -\infty \wedge w < \infty \wedge w \neq 0 \\\frac{t_{2}}{2} & \text{otherwise} \end{cases} 
t2/2 + Piecewise((-i/(4*w) + i*exp(-2*i*t2*w)/(4*w), (w > -oo)∧(w < oo)∧(Ne(w, 0))), (t2/2, True))

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

    © Sr Examen – Калькулятор