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Integral de 2\π*x*(π+x)*sin(n*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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