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Resolución de integrales/ sin(n*x)/ x^2*sin(n*x)

Integral de x^2*sin(n*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1               
  /               
 |                
 |   2            
 |  x *sin(n*x) dx
 |                
/                 
0
$$\int\limits_{0}^{1} x^{2} \sin{\left(n x \right)}\, dx$$ 
Integral(x^2*sin(n*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
                          //                  0                     for n = 0\                              
                          ||                                                 |                              
                          || //cos(n*x)   x*sin(n*x)            \            |                              
  /                       || ||-------- + ----------  for n != 0|            |                              
 |                        || ||    2          n                 |            |      //    0       for n = 0\
 |  2                     || ||   n                             |            |    2 ||                     |
 | x *sin(n*x) dx = C - 2*|<-|<                                 |            | + x *|<-cos(n*x)            |
 |                        || ||          2                      |            |      ||----------  otherwise|
/                         || ||         x                       |            |      \\    n                /
                          || ||         --            otherwise |            |                              
                          || \\         2                       /            |                              
                          ||--------------------------------------  otherwise|                              
                          \\                  n                              /
$$\int x^{2} \sin{\left(n x \right)}\, dx = C + x^{2} \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\cos{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right) - 2 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\begin{cases} \frac{x \sin{\left(n x \right)}}{n} + \frac{\cos{\left(n x \right)}}{n^{2}} & \text{for}\: n \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}\right)$$
Simplificar
Respuesta [src] 
/  2    cos(n)   2*cos(n)   2*sin(n)                                  
|- -- - ------ + -------- + --------  for And(n > -oo, n < oo, n != 0)
|   3     n          3          2                                     
<  n                n          n                                      
|                                                                     
|                 0                              otherwise            
\
$$\begin{cases} - \frac{\cos{\left(n \right)}}{n} + \frac{2 \sin{\left(n \right)}}{n^{2}} + \frac{2 \cos{\left(n \right)}}{n^{3}} - \frac{2}{n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$ 
=
= 
/  2    cos(n)   2*cos(n)   2*sin(n)                                  
|- -- - ------ + -------- + --------  for And(n > -oo, n < oo, n != 0)
|   3     n          3          2                                     
<  n                n          n                                      
|                                                                     
|                 0                              otherwise            
\
$$\begin{cases} - \frac{\cos{\left(n \right)}}{n} + \frac{2 \sin{\left(n \right)}}{n^{2}} + \frac{2 \cos{\left(n \right)}}{n^{3}} - \frac{2}{n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$ 
Piecewise((-2/n^3 - cos(n)/n + 2*cos(n)/n^3 + 2*sin(n)/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.

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