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Resolución de integrales/ i*n*x/ x^3*sin(pi*n*x/l)

Integral de x^3*sin(pi*n*x/l) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  l                  
  /                  
 |                   
 |   3    /pi*n*x\   
 |  x *sin|------| dx
 |        \  l   /   
 |                   
/                    
0
$$\int\limits_{0}^{l} x^{3} \sin{\left(\frac{x \pi n}{l} \right)}\, dx$$ 
Integral(x^3*sin(((pi*n)*x)/l), (x, 0, l))
Respuesta (Indefinida) [src] 
                             //                                      0                                        for n = 0\                                   
                             ||                                                                                        |                                   
                             ||   //     3    /pi*n*x\      2    /pi*n*x\        2    /pi*n*x\            \            |                                   
                             ||   ||  2*l *sin|------|   l*x *sin|------|   2*x*l *cos|------|            |            |                                   
  /                          ||   ||          \  l   /           \  l   /             \  l   /            |            |      //       0         for n = 0\
 |                           ||   ||- ---------------- + ---------------- + ------------------  for n != 0|            |      ||                          |
 |  3    /pi*n*x\            ||   ||         3  3              pi*n                 2  2                  |            |    3 ||      /pi*n*x\            |
 | x *sin|------| dx = C - 3*|<-l*|<       pi *n                                  pi *n                   |            | + x *|<-l*cos|------|            |
 |       \  l   /            ||   ||                                                                      |            |      ||      \  l   /            |
 |                           ||   ||                             3                                        |            |      ||---------------  otherwise|
/                            ||   ||                            x                                         |            |      \\      pi*n                /
                             ||   ||                            --                              otherwise |            |                                   
                             ||   \\                            3                                         /            |                                   
                             ||-----------------------------------------------------------------------------  otherwise|                                   
                             \\                                     pi*n                                               /
$$\int x^{3} \sin{\left(\frac{x \pi n}{l} \right)}\, dx = C + x^{3} \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{l \cos{\left(\frac{\pi n x}{l} \right)}}{\pi n} & \text{otherwise} \end{cases}\right) - 3 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{l \left(\begin{cases} - \frac{2 l^{3} \sin{\left(\frac{\pi n x}{l} \right)}}{\pi^{3} n^{3}} + \frac{2 l^{2} x \cos{\left(\frac{\pi n x}{l} \right)}}{\pi^{2} n^{2}} + \frac{l x^{2} \sin{\left(\frac{\pi n x}{l} \right)}}{\pi n} & \text{for}\: n \neq 0 \\\frac{x^{3}}{3} & \text{otherwise} \end{cases}\right)}{\pi n} & \text{otherwise} \end{cases}\right)$$
Simplificar
Respuesta [src] 
/   4                4                4                4                                            
|  l *cos(pi*n)   6*l *sin(pi*n)   3*l *sin(pi*n)   6*l *cos(pi*n)                                  
|- ------------ - -------------- + -------------- + --------------  for And(n > -oo, n < oo, n != 0)
<      pi*n             4  4             2  2             3  3                                      
|                     pi *n            pi *n            pi *n                                       
|                                                                                                   
\                                0                                             otherwise
$$\begin{cases} - \frac{l^{4} \cos{\left(\pi n \right)}}{\pi n} + \frac{3 l^{4} \sin{\left(\pi n \right)}}{\pi^{2} n^{2}} + \frac{6 l^{4} \cos{\left(\pi n \right)}}{\pi^{3} n^{3}} - \frac{6 l^{4} \sin{\left(\pi n \right)}}{\pi^{4} n^{4}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$ 
=
= 
/   4                4                4                4                                            
|  l *cos(pi*n)   6*l *sin(pi*n)   3*l *sin(pi*n)   6*l *cos(pi*n)                                  
|- ------------ - -------------- + -------------- + --------------  for And(n > -oo, n < oo, n != 0)
<      pi*n             4  4             2  2             3  3                                      
|                     pi *n            pi *n            pi *n                                       
|                                                                                                   
\                                0                                             otherwise
$$\begin{cases} - \frac{l^{4} \cos{\left(\pi n \right)}}{\pi n} + \frac{3 l^{4} \sin{\left(\pi n \right)}}{\pi^{2} n^{2}} + \frac{6 l^{4} \cos{\left(\pi n \right)}}{\pi^{3} n^{3}} - \frac{6 l^{4} \sin{\left(\pi n \right)}}{\pi^{4} n^{4}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$ 
Piecewise((-l^4*cos(pi*n)/(pi*n) - 6*l^4*sin(pi*n)/(pi^4*n^4) + 3*l^4*sin(pi*n)/(pi^2*n^2) + 6*l^4*cos(pi*n)/(pi^3*n^3), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (0, True))

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

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