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Resolución de integrales/ (x^2)/ cos(n*x)/ -4*(x^2)*cos(n*x)

Integral de -4*(x^2)*cos(n*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 -1                  
  /                  
 |                   
 |      2            
 |  -4*x *cos(n*x) dx
 |                   
/                    
-pi
$$\int\limits_{- \pi}^{-1} - 4 x^{2} \cos{\left(n x \right)}\, dx$$ 
Integral((-4*x^2)*cos(n*x), (x, -pi, -1))
Respuesta (Indefinida) [src] 
                             //                 3                           \                              
                             ||                x                            |                              
                             ||                --                  for n = 0|                              
                             ||                3                            |                              
  /                          ||                                             |                              
 |                           ||/sin(n*x)   x*cos(n*x)                       |        //   x      for n = 0\
 |     2                     |||-------- - ----------  for n != 0           |      2 ||                   |
 | -4*x *cos(n*x) dx = C + 8*|<|    2          n                            | - 4*x *|
$$\int - 4 x^{2} \cos{\left(n x \right)}\, dx = C - 4 x^{2} \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right) + 8 \left(\begin{cases} \frac{x^{3}}{3} & \text{for}\: n = 0 \\\frac{\begin{cases} - \frac{x \cos{\left(n x \right)}}{n} + \frac{\sin{\left(n x \right)}}{n^{2}} & \text{for}\: n \neq 0 \\0 & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}\right)$$
Simplificar
Respuesta [src] 
/                                                                      2                                            
|  8*sin(n)   4*sin(n)   8*sin(pi*n)   8*cos(n)   8*pi*cos(pi*n)   4*pi *sin(pi*n)                                  
|- -------- + -------- + ----------- + -------- - -------------- - ---------------  for And(n > -oo, n < oo, n != 0)
|      3         n             3           2             2                n                                         
|     n                       n           n             n                                                           
<                                                                                                                   
|                                            3                                                                      
|                                    4   4*pi                                                                       
|                                    - - -----                                                 otherwise            
|                                    3     3                                                                        
\
$$\begin{cases} \frac{4 \sin{\left(n \right)}}{n} - \frac{4 \pi^{2} \sin{\left(\pi n \right)}}{n} + \frac{8 \cos{\left(n \right)}}{n^{2}} - \frac{8 \pi \cos{\left(\pi n \right)}}{n^{2}} - \frac{8 \sin{\left(n \right)}}{n^{3}} + \frac{8 \sin{\left(\pi n \right)}}{n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{4}{3} - \frac{4 \pi^{3}}{3} & \text{otherwise} \end{cases}$$ 
=
= 
/                                                                      2                                            
|  8*sin(n)   4*sin(n)   8*sin(pi*n)   8*cos(n)   8*pi*cos(pi*n)   4*pi *sin(pi*n)                                  
|- -------- + -------- + ----------- + -------- - -------------- - ---------------  for And(n > -oo, n < oo, n != 0)
|      3         n             3           2             2                n                                         
|     n                       n           n             n                                                           
<                                                                                                                   
|                                            3                                                                      
|                                    4   4*pi                                                                       
|                                    - - -----                                                 otherwise            
|                                    3     3                                                                        
\
$$\begin{cases} \frac{4 \sin{\left(n \right)}}{n} - \frac{4 \pi^{2} \sin{\left(\pi n \right)}}{n} + \frac{8 \cos{\left(n \right)}}{n^{2}} - \frac{8 \pi \cos{\left(\pi n \right)}}{n^{2}} - \frac{8 \sin{\left(n \right)}}{n^{3}} + \frac{8 \sin{\left(\pi n \right)}}{n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{4}{3} - \frac{4 \pi^{3}}{3} & \text{otherwise} \end{cases}$$ 
Piecewise((-8*sin(n)/n^3 + 4*sin(n)/n + 8*sin(pi*n)/n^3 + 8*cos(n)/n^2 - 8*pi*cos(pi*n)/n^2 - 4*pi^2*sin(pi*n)/n, (n > -oo)∧(n < oo)∧(Ne(n, 0))), (4/3 - 4*pi^3/3, True))

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

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