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Resolución de integrales/ (-x-1)/ (-x-1)sin(pi*x*n)

Integral de (-x-1)sin(pi*x*n) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  0                        
  /                        
 |                         
 |  (-x - 1)*sin(pi*x*n) dx
 |                         
/                          
-1
∫−10(−x−1)sin⁡(nπx) dx\int\limits_{-1}^{0} \left(- x - 1\right) \sin{\left(n \pi x \right)}\, dx 
Integral((-x - 1)*sin((pi*x)*n), (x, -1, 0))
Respuesta (Indefinida) [src] 
                                                                                               //               0                 for n = 0\
                                                                                               ||                                          |
  /                              //      0        for n = 0\     //      0        for n = 0\   || //sin(pi*n*x)               \            |
 |                               ||                        |     ||                        |   || ||-----------  for pi*n != 0|            |
 | (-x - 1)*sin(pi*x*n) dx = C - |<-cos(pi*n*x)            | - x*|<-cos(pi*n*x)            | + |<-|<    pi*n                  |            |
 |                               ||-------------  otherwise|     ||-------------  otherwise|   || ||                          |            |
/                                \\     pi*n               /     \\     pi*n               /   || \\     x         otherwise  /            |
                                                                                               ||-------------------------------  otherwise|
                                                                                               \\              pi*n                        /
∫(−x−1)sin⁡(nπx) dx=C−x({0for n=0−cos⁡(πnx)πnotherwise)+{0for n=0−{sin⁡(πnx)πnfor πn≠0xotherwiseπnotherwise−{0for n=0−cos⁡(πnx)πnotherwise\int \left(- x - 1\right) \sin{\left(n \pi x \right)}\, dx = C - x \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\cos{\left(\pi n x \right)}}{\pi n} & \text{otherwise} \end{cases}\right) + \begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\begin{cases} \frac{\sin{\left(\pi n x \right)}}{\pi n} & \text{for}\: \pi n \neq 0 \\x & \text{otherwise} \end{cases}}{\pi n} & \text{otherwise} \end{cases} - \begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\cos{\left(\pi n x \right)}}{\pi n} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/ 1     sin(pi*n)                                  
|---- - ---------  for And(n > -oo, n < oo, n != 0)
|pi*n       2  2                                   
<         pi *n                                    
|                                                  
|       0                     otherwise            
\
{1πn−sin⁡(πn)π2n2for n>−∞∧n<∞∧n≠00otherwise\begin{cases} \frac{1}{\pi n} - \frac{\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} 
=
= 
/ 1     sin(pi*n)                                  
|---- - ---------  for And(n > -oo, n < oo, n != 0)
|pi*n       2  2                                   
<         pi *n                                    
|                                                  
|       0                     otherwise            
\
{1πn−sin⁡(πn)π2n2for n>−∞∧n<∞∧n≠00otherwise\begin{cases} \frac{1}{\pi n} - \frac{\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((1/(pi*n) - 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.

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