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

Integral de x*(x-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\   
 |  x*(x - 2)*cos|------| dx
 |               \  2   /   
 |                          
/                           
0
∫02x(x−2)cos⁡(xπn2) dx\int\limits_{0}^{2} x \left(x - 2\right) \cos{\left(\frac{x \pi n}{2} \right)}\, dx 
Integral((x*(x - 2))*cos(((pi*n)*x)/2), (x, 0, 2))
Respuesta (Indefinida) [src] 
                                    //                        3                                  \                                                                                                                        
                                    ||                       x                                   |     //                 2                           \                                                                   
                                    ||                       --                         for n = 0|     ||                x                            |                                                                   
                                    ||                       3                                   |     ||                --                  for n = 0|                                                                   
                                    ||                                                           |     ||                2                            |                                                                   
  /                                 ||  //     /pi*n*x\          /pi*n*x\            \           |     ||                                             |      //      x        for n = 0\       //      x        for n = 0\
 |                                  ||  ||4*sin|------|   2*x*cos|------|            |           |     ||  //      /pi*n*x\               \           |      ||                        |       ||                        |
 |              /pi*n*x\            ||  ||     \  2   /          \  2   /            |           |     ||  ||-2*cos|------|               |           |    2 ||     /pi*n*x\           |       ||     /pi*n*x\           |
 | x*(x - 2)*cos|------| dx = C - 2*|<  ||------------- - ---------------  for n != 0|           | + 2*|<  ||      \  2   /      pi*n     |           | + x *|<2*sin|------|           | - 2*x*|<2*sin|------|           |
 |              \  2   /            ||2*|<      2  2            pi*n                 |           |     ||2*|<--------------  for ---- != 0|           |      ||     \  2   /           |       ||     \  2   /           |
 |                                  ||  ||    pi *n                                  |           |     ||  ||     pi*n            2       |           |      ||-------------  otherwise|       ||-------------  otherwise|
/                                   ||  ||                                           |           |     ||  ||                             |           |      \\     pi*n               /       \\     pi*n               /
                                    ||  ||               0                 otherwise |           |     ||  \\      0           otherwise  /           |                                                                   
                                    ||  \\                                           /           |     ||----------------------------------  otherwise|                                                                   
                                    ||------------------------------------------------  otherwise|     ||               pi*n                          |                                                                   
                                    ||                      pi*n                                 |     \\                                             /                                                                   
                                    \\                                                           /
∫x(x−2)cos⁡(xπn2) dx=C+x2({xfor n=02sin⁡(πnx2)πnotherwise)−2x({xfor n=02sin⁡(πnx2)πnotherwise)+2({x22for n=02({−2cos⁡(πnx2)πnfor πn2≠00otherwise)πnotherwise)−2({x33for n=02({−2xcos⁡(πnx2)πn+4sin⁡(πnx2)π2n2for n≠00otherwise)πnotherwise)\int x \left(x - 2\right) \cos{\left(\frac{x \pi n}{2} \right)}\, dx = C + x^{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) - 2 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} \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) - 2 \left(\begin{cases} \frac{x^{3}}{3} & \text{for}\: n = 0 \\\frac{2 \left(\begin{cases} - \frac{2 x \cos{\left(\frac{\pi n x}{2} \right)}}{\pi n} + \frac{4 \sin{\left(\frac{\pi n x}{2} \right)}}{\pi^{2} n^{2}} & \text{for}\: n \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{\pi n} & \text{otherwise} \end{cases}\right)
Simplificar
Respuesta [src] 
/  8      16*sin(pi*n)   8*cos(pi*n)                                  
|------ - ------------ + -----------  for And(n > -oo, n < oo, n != 0)
|  2  2        3  3           2  2
{8cos⁡(πn)π2n2+8π2n2−16sin⁡(πn)π3n3for n>−∞∧n<∞∧n≠0−43otherwise\begin{cases} \frac{8 \cos{\left(\pi n \right)}}{\pi^{2} n^{2}} + \frac{8}{\pi^{2} n^{2}} - \frac{16 \sin{\left(\pi n \right)}}{\pi^{3} n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\- \frac{4}{3} & \text{otherwise} \end{cases} 
=
= 
/  8      16*sin(pi*n)   8*cos(pi*n)                                  
|------ - ------------ + -----------  for And(n > -oo, n < oo, n != 0)
|  2  2        3  3           2  2
{8cos⁡(πn)π2n2+8π2n2−16sin⁡(πn)π3n3for n>−∞∧n<∞∧n≠0−43otherwise\begin{cases} \frac{8 \cos{\left(\pi n \right)}}{\pi^{2} n^{2}} + \frac{8}{\pi^{2} n^{2}} - \frac{16 \sin{\left(\pi n \right)}}{\pi^{3} n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\- \frac{4}{3} & \text{otherwise} \end{cases} 
Piecewise((8/(pi^2*n^2) - 16*sin(pi*n)/(pi^3*n^3) + 8*cos(pi*n)/(pi^2*n^2), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (-4/3, True))

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

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