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

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

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

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