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Resolución de integrales/ e^x-1/ (e^x-1)e^(-i*w*n*x)

Integral de (e^x-1)e^(-i*w*n*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                      
  /                      
 |                       
 |  / x    \  -I*w*n*x   
 |  \E  - 1/*E         dx
 |                       
/                        
0
$$\int\limits_{0}^{1} e^{x n - i w} \left(e^{x} - 1\right)\, dx$$ 
Integral((E^x - 1)*E^((((-i)*w)*n)*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                            //     x       for Or(n = 0, w = 0)\   //                           x                                                   x                                   \
 |                             ||                                 |   ||                        I*e                                               n*w*e                                 -I |
 | / x    \  -I*w*n*x          ||   -I*w*n*x                      |   ||- ------------------------------------------------ - ------------------------------------------------  for n != ---|
 | \E  - 1/*E         dx = C - |
$$\int e^{x n - i w} \left(e^{x} - 1\right)\, dx = C - \begin{cases} x & \text{for}\: n = 0 \vee w = 0 \\\frac{i e^{x n - i w}}{n w} & \text{otherwise} \end{cases} + \begin{cases} - \frac{n w e^{x}}{i n^{2} w^{2} e^{i n w x} - 2 n w e^{i n w x} - i e^{i n w x}} - \frac{i e^{x}}{i n^{2} w^{2} e^{i n w x} - 2 n w e^{i n w x} - i e^{i n w x}} & \text{for}\: n \neq - \frac{i}{w} \\x & \text{otherwise} \end{cases}$$
Simplificar
Respuesta [src] 
/                                                                                                                                                                                                                                                                                                    /   /           -I \                        /           -I        \              \
|                                                                                                                                           -2 + E                                                                                                                                             for Or|And|n = 0, n = ---|, And(n = 0, w = 0), And|n = 0, n = ---, w = 0|, n = 0, w = 0|
|                                                                                                                                                                                                                                                                                                    \   \            w /                        \            w        /              /
|                                                                                                                                                                                                                                                                                                                                                                                      
|                                                                                                                                             -1                                                                                                                                                                               /   /    -I        \      -I \                          
|                                                                                                                                            e                                                                                                                                                                           for Or|And|n = ---, w = 0|, n = ---|                          
<                                                                                                                                                                                                                                                                                                                              \   \     w        /       w /                          
|                                                                                                                                                                                                                                                                                                                                                                                      
|                                                                                                                                                                                                                      2  2                                            2  2                                                                                                            
|                     I                                    I                        n*w                                2*n*w                                          E*n*w                                         I*n *w                                        E*I*n *w                                                                                                             
|-------------------------------------------- - ----------------------- - ----------------------- + -------------------------------------------- - -------------------------------------------- - -------------------------------------------- + --------------------------------------------                                         otherwise                                        
| 3  3  I*n*w        I*n*w        2  2  I*n*w    3  3              2  2    3  3              2  2    3  3  I*n*w        I*n*w        2  2  I*n*w    3  3  I*n*w        I*n*w        2  2  I*n*w    3  3  I*n*w        I*n*w        2  2  I*n*w    3  3  I*n*w        I*n*w        2  2  I*n*w                                                                                          
\n *w *e      - n*w*e      + 2*I*n *w *e        n *w  - n*w + 2*I*n *w    n *w  - n*w + 2*I*n *w    n *w *e      - n*w*e      + 2*I*n *w *e        n *w *e      - n*w*e      + 2*I*n *w *e        n *w *e      - n*w*e      + 2*I*n *w *e        n *w *e      - n*w*e      + 2*I*n *w *e
$$\begin{cases} -2 + e & \text{for}\: \left(n = 0 \wedge n = - \frac{i}{w}\right) \vee \left(n = 0 \wedge w = 0\right) \vee \left(n = 0 \wedge n = - \frac{i}{w} \wedge w = 0\right) \vee n = 0 \vee w = 0 \\e^{-1} & \text{for}\: \left(n = - \frac{i}{w} \wedge w = 0\right) \vee n = - \frac{i}{w} \\- \frac{i n^{2} w^{2}}{n^{3} w^{3} e^{i n w} + 2 i n^{2} w^{2} e^{i n w} - n w e^{i n w}} + \frac{e i n^{2} w^{2}}{n^{3} w^{3} e^{i n w} + 2 i n^{2} w^{2} e^{i n w} - n w e^{i n w}} - \frac{e n w}{n^{3} w^{3} e^{i n w} + 2 i n^{2} w^{2} e^{i n w} - n w e^{i n w}} + \frac{2 n w}{n^{3} w^{3} e^{i n w} + 2 i n^{2} w^{2} e^{i n w} - n w e^{i n w}} - \frac{n w}{n^{3} w^{3} + 2 i n^{2} w^{2} - n w} + \frac{i}{n^{3} w^{3} e^{i n w} + 2 i n^{2} w^{2} e^{i n w} - n w e^{i n w}} - \frac{i}{n^{3} w^{3} + 2 i n^{2} w^{2} - n w} & \text{otherwise} \end{cases}$$ 
=
= 
/                                                                                                                                                                                                                                                                                                    /   /           -I \                        /           -I        \              \
|                                                                                                                                           -2 + E                                                                                                                                             for Or|And|n = 0, n = ---|, And(n = 0, w = 0), And|n = 0, n = ---, w = 0|, n = 0, w = 0|
|                                                                                                                                                                                                                                                                                                    \   \            w /                        \            w        /              /
|                                                                                                                                                                                                                                                                                                                                                                                      
|                                                                                                                                             -1                                                                                                                                                                               /   /    -I        \      -I \                          
|                                                                                                                                            e                                                                                                                                                                           for Or|And|n = ---, w = 0|, n = ---|                          
<                                                                                                                                                                                                                                                                                                                              \   \     w        /       w /                          
|                                                                                                                                                                                                                                                                                                                                                                                      
|                                                                                                                                                                                                                      2  2                                            2  2                                                                                                            
|                     I                                    I                        n*w                                2*n*w                                          E*n*w                                         I*n *w                                        E*I*n *w                                                                                                             
|-------------------------------------------- - ----------------------- - ----------------------- + -------------------------------------------- - -------------------------------------------- - -------------------------------------------- + --------------------------------------------                                         otherwise                                        
| 3  3  I*n*w        I*n*w        2  2  I*n*w    3  3              2  2    3  3              2  2    3  3  I*n*w        I*n*w        2  2  I*n*w    3  3  I*n*w        I*n*w        2  2  I*n*w    3  3  I*n*w        I*n*w        2  2  I*n*w    3  3  I*n*w        I*n*w        2  2  I*n*w                                                                                          
\n *w *e      - n*w*e      + 2*I*n *w *e        n *w  - n*w + 2*I*n *w    n *w  - n*w + 2*I*n *w    n *w *e      - n*w*e      + 2*I*n *w *e        n *w *e      - n*w*e      + 2*I*n *w *e        n *w *e      - n*w*e      + 2*I*n *w *e        n *w *e      - n*w*e      + 2*I*n *w *e
$$\begin{cases} -2 + e & \text{for}\: \left(n = 0 \wedge n = - \frac{i}{w}\right) \vee \left(n = 0 \wedge w = 0\right) \vee \left(n = 0 \wedge n = - \frac{i}{w} \wedge w = 0\right) \vee n = 0 \vee w = 0 \\e^{-1} & \text{for}\: \left(n = - \frac{i}{w} \wedge w = 0\right) \vee n = - \frac{i}{w} \\- \frac{i n^{2} w^{2}}{n^{3} w^{3} e^{i n w} + 2 i n^{2} w^{2} e^{i n w} - n w e^{i n w}} + \frac{e i n^{2} w^{2}}{n^{3} w^{3} e^{i n w} + 2 i n^{2} w^{2} e^{i n w} - n w e^{i n w}} - \frac{e n w}{n^{3} w^{3} e^{i n w} + 2 i n^{2} w^{2} e^{i n w} - n w e^{i n w}} + \frac{2 n w}{n^{3} w^{3} e^{i n w} + 2 i n^{2} w^{2} e^{i n w} - n w e^{i n w}} - \frac{n w}{n^{3} w^{3} + 2 i n^{2} w^{2} - n w} + \frac{i}{n^{3} w^{3} e^{i n w} + 2 i n^{2} w^{2} e^{i n w} - n w e^{i n w}} - \frac{i}{n^{3} w^{3} + 2 i n^{2} w^{2} - n w} & \text{otherwise} \end{cases}$$ 
Piecewise((-2 + E, (n = 0)∨(w = 0)∨((n = 0)∧(w = 0))∨((n = 0)∧(n = -i/w))∨((n = 0)∧(w = 0)∧(n = -i/w))), (exp(-1), (n = -i/w)∨((w = 0)∧(n = -i/w))), (i/(n^3*w^3*exp(i*n*w) - n*w*exp(i*n*w) + 2*i*n^2*w^2*exp(i*n*w)) - i/(n^3*w^3 - n*w + 2*i*n^2*w^2) - n*w/(n^3*w^3 - n*w + 2*i*n^2*w^2) + 2*n*w/(n^3*w^3*exp(i*n*w) - n*w*exp(i*n*w) + 2*i*n^2*w^2*exp(i*n*w)) - E*n*w/(n^3*w^3*exp(i*n*w) - n*w*exp(i*n*w) + 2*i*n^2*w^2*exp(i*n*w)) - i*n^2*w^2/(n^3*w^3*exp(i*n*w) - n*w*exp(i*n*w) + 2*i*n^2*w^2*exp(i*n*w)) + E*i*n^2*w^2/(n^3*w^3*exp(i*n*w) - n*w*exp(i*n*w) + 2*i*n^2*w^2*exp(i*n*w)), True))

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

    © Sr Examen – Калькулятор