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Integral de (x)*sin((pi*n*x)/4) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
  4                 
  /                 
 |                  
 |       /pi*n*x\   
 |  x*sin|------| dx
 |       \  4   /   
 |                  
/                   
0                   
$$\int\limits_{0}^{4} x \sin{\left(\frac{x \pi n}{4} \right)}\, dx$$
Integral(x*sin(((pi*n)*x)/4), (x, 0, 4))
Respuesta (Indefinida) [src]
                          //                0                   for n = 0\                                 
                          ||                                             |                                 
  /                       ||   //     /pi*n*x\               \           |     //      0         for n = 0\
 |                        ||   ||4*sin|------|               |           |     ||                         |
 |      /pi*n*x\          ||   ||     \  4   /      pi*n     |           |     ||      /pi*n*x\           |
 | x*sin|------| dx = C - |<-4*|<-------------  for ---- != 0|           | + x*|<-4*cos|------|           |
 |      \  4   /          ||   ||     pi*n           4       |           |     ||      \  4   /           |
 |                        ||   ||                            |           |     ||--------------  otherwise|
/                         ||   \\      x          otherwise  /           |     \\     pi*n                /
                          ||----------------------------------  otherwise|                                 
                          \\               pi*n                          /                                 
$$\int x \sin{\left(\frac{x \pi n}{4} \right)}\, dx = C + x \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{4 \cos{\left(\frac{\pi n x}{4} \right)}}{\pi n} & \text{otherwise} \end{cases}\right) - \begin{cases} 0 & \text{for}\: n = 0 \\- \frac{4 \left(\begin{cases} \frac{4 \sin{\left(\frac{\pi n x}{4} \right)}}{\pi n} & \text{for}\: \frac{\pi n}{4} \neq 0 \\x & \text{otherwise} \end{cases}\right)}{\pi n} & \text{otherwise} \end{cases}$$
Respuesta [src]
/  16*cos(pi*n)   16*sin(pi*n)                                  
|- ------------ + ------------  for And(n > -oo, n < oo, n != 0)
|      pi*n            2  2                                     
<                    pi *n                                      
|                                                               
|              0                           otherwise            
\                                                               
$$\begin{cases} - \frac{16 \cos{\left(\pi n \right)}}{\pi n} + \frac{16 \sin{\left(\pi n \right)}}{\pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$
=
=
/  16*cos(pi*n)   16*sin(pi*n)                                  
|- ------------ + ------------  for And(n > -oo, n < oo, n != 0)
|      pi*n            2  2                                     
<                    pi *n                                      
|                                                               
|              0                           otherwise            
\                                                               
$$\begin{cases} - \frac{16 \cos{\left(\pi n \right)}}{\pi n} + \frac{16 \sin{\left(\pi n \right)}}{\pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$
Piecewise((-16*cos(pi*n)/(pi*n) + 16*sin(pi*n)/(pi^2*n^2), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (0, True))

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