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Resolución de integrales/ i*n*x/ (-5x-2)*cos(pi*n*x/2)

Integral de (-5x-2)*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\   
 |  (-5*x - 2)*cos|------| dx
 |                \  2   /   
 |                           
/                            
0
$$\int\limits_{0}^{2} \left(- 5 x - 2\right) \cos{\left(\frac{x \pi n}{2} \right)}\, dx$$ 
Integral((-5*x - 2)*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\           |
 | (-5*x - 2)*cos|------| dx = C - 2*|<2*sin|------|           | + 5*|<  ||      \  2   /      pi*n     |           | - 5*x*|<2*sin|------|           |
 |               \  2   /            ||     \  2   /           |     ||2*|<--------------  for ---- != 0|           |       ||     \  2   /           |
 |                                   ||-------------  otherwise|     ||  ||     pi*n            2       |           |       ||-------------  otherwise|
/                                    \\     pi*n               /     ||  ||                             |           |       \\     pi*n               /
                                                                     ||  \\      0           otherwise  /           |                                  
                                                                     ||----------------------------------  otherwise|                                  
                                                                     ||               pi*n                          |                                  
                                                                     \\                                             /
$$\int \left(- 5 x - 2\right) \cos{\left(\frac{x \pi n}{2} \right)}\, dx = C - 5 x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{2 \sin{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{otherwise} \end{cases}\right) - 2 \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{2 \sin{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{otherwise} \end{cases}\right) + 5 \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{2 \left(\begin{cases} - \frac{2 \cos{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{for}\: \frac{\pi n}{2} \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{\pi n} & \text{otherwise} \end{cases}\right)$$
Simplificar
Respuesta [src] 
/  20     24*sin(pi*n)   20*cos(pi*n)                                  
|------ - ------------ - ------------  for And(n > -oo, n < oo, n != 0)
|  2  2       pi*n            2  2
$$\begin{cases} - \frac{24 \sin{\left(\pi n \right)}}{\pi n} - \frac{20 \cos{\left(\pi n \right)}}{\pi^{2} n^{2}} + \frac{20}{\pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\-14 & \text{otherwise} \end{cases}$$ 
=
= 
/  20     24*sin(pi*n)   20*cos(pi*n)                                  
|------ - ------------ - ------------  for And(n > -oo, n < oo, n != 0)
|  2  2       pi*n            2  2
$$\begin{cases} - \frac{24 \sin{\left(\pi n \right)}}{\pi n} - \frac{20 \cos{\left(\pi n \right)}}{\pi^{2} n^{2}} + \frac{20}{\pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\-14 & \text{otherwise} \end{cases}$$ 
Piecewise((20/(pi^2*n^2) - 24*sin(pi*n)/(pi*n) - 20*cos(pi*n)/(pi^2*n^2), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (-14, True))

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

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