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Resolución de integrales/ i*n*x/ (x-1)/ (x-1)*cos((pi*n*x)/2)

Integral de (x-1)*cos((pi*n*x)/2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 -2                       
  /                       
 |                        
 |             /pi*n*x\   
 |  (x - 1)*cos|------| dx
 |             \  2   /   
 |                        
/                         
0
$$\int\limits_{0}^{-2} \left(x - 1\right) \cos{\left(\frac{x \pi n}{2} \right)}\, dx$$ 
Integral((x - 1)*cos(((pi*n)*x)/2), (x, 0, -2))
Respuesta (Indefinida) [src] 
                                                              //                 2                           \                                
                                                              ||                x                            |                                
                                                              ||                --                  for n = 0|                                
                                                              ||                2                            |                                
  /                             //      x        for n = 0\   ||                                             |     //      x        for n = 0\
 |                              ||                        |   ||  //      /pi*n*x\               \           |     ||                        |
 |            /pi*n*x\          ||     /pi*n*x\           |   ||  ||-2*cos|------|               |           |     ||     /pi*n*x\           |
 | (x - 1)*cos|------| dx = C - |<2*sin|------|           | - |<  ||      \  2   /      pi*n     |           | + x*|<2*sin|------|           |
 |            \  2   /          ||     \  2   /           |   ||2*|<--------------  for ---- != 0|           |     ||     \  2   /           |
 |                              ||-------------  otherwise|   ||  ||     pi*n            2       |           |     ||-------------  otherwise|
/                               \\     pi*n               /   ||  ||                             |           |     \\     pi*n               /
                                                              ||  \\      0           otherwise  /           |                                
                                                              ||----------------------------------  otherwise|                                
                                                              ||               pi*n                          |                                
                                                              \\                                             /
$$\int \left(x - 1\right) \cos{\left(\frac{x \pi n}{2} \right)}\, dx = C + x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{2 \sin{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{otherwise} \end{cases}\right) - \begin{cases} x & \text{for}\: n = 0 \\\frac{2 \sin{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{otherwise} \end{cases} - \begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{2 \left(\begin{cases} - \frac{2 \cos{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{for}\: \frac{\pi n}{2} \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{\pi n} & \text{otherwise} \end{cases}$$
Simplificar
Respuesta [src] 
/    4      4*cos(pi*n)   6*sin(pi*n)                                  
|- ------ + ----------- + -----------  for And(n > -oo, n < oo, n != 0)
|    2  2        2  2         pi*n                                     
<  pi *n       pi *n                                                   
|                                                                      
|                 4                               otherwise            
\
$$\begin{cases} \frac{6 \sin{\left(\pi n \right)}}{\pi n} + \frac{4 \cos{\left(\pi n \right)}}{\pi^{2} n^{2}} - \frac{4}{\pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\4 & \text{otherwise} \end{cases}$$ 
=
= 
/    4      4*cos(pi*n)   6*sin(pi*n)                                  
|- ------ + ----------- + -----------  for And(n > -oo, n < oo, n != 0)
|    2  2        2  2         pi*n                                     
<  pi *n       pi *n                                                   
|                                                                      
|                 4                               otherwise            
\
$$\begin{cases} \frac{6 \sin{\left(\pi n \right)}}{\pi n} + \frac{4 \cos{\left(\pi n \right)}}{\pi^{2} n^{2}} - \frac{4}{\pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\4 & \text{otherwise} \end{cases}$$ 
Piecewise((-4/(pi^2*n^2) + 4*cos(pi*n)/(pi^2*n^2) + 6*sin(pi*n)/(pi*n), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (4, True))

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

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