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Resolución de integrales/ i*n*x/ (3*x-1)*sin(pi*n*x)

Integral de (3*x-1)*sin(pi*n*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  0                         
  /                         
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 |  (3*x - 1)*sin(pi*n*x) dx
 |                          
/                           
-1
∫−10(3x−1)sin⁡(xπn) dx\int\limits_{-1}^{0} \left(3 x - 1\right) \sin{\left(x \pi n \right)}\, dx 
Integral((3*x - 1)*sin((pi*n)*x), (x, -1, 0))
Respuesta (Indefinida) [src] 
                                                                  //               0                 for n = 0\                                  
                                                                  ||                                          |                                  
  /                               //      0        for n = 0\     || //sin(pi*n*x)               \            |       //      0        for n = 0\
 |                                ||                        |     || ||-----------  for pi*n != 0|            |       ||                        |
 | (3*x - 1)*sin(pi*n*x) dx = C - |<-cos(pi*n*x)            | - 3*|<-|<    pi*n                  |            | + 3*x*|<-cos(pi*n*x)            |
 |                                ||-------------  otherwise|     || ||                          |            |       ||-------------  otherwise|
/                                 \\     pi*n               /     || \\     x         otherwise  /            |       \\     pi*n               /
                                                                  ||-------------------------------  otherwise|                                  
                                                                  \\              pi*n                        /
∫(3x−1)sin⁡(xπn) dx=C+3x({0for n=0−cos⁡(πnx)πnotherwise)−3({0for n=0−{sin⁡(πnx)πnfor πn≠0xotherwiseπnotherwise)−{0for n=0−cos⁡(πnx)πnotherwise\int \left(3 x - 1\right) \sin{\left(x \pi n \right)}\, dx = C + 3 x \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\cos{\left(\pi n x \right)}}{\pi n} & \text{otherwise} \end{cases}\right) - 3 \left(\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}\right) - \begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\cos{\left(\pi n x \right)}}{\pi n} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/ 1     4*cos(pi*n)   3*sin(pi*n)                                  
|---- - ----------- + -----------  for And(n > -oo, n < oo, n != 0)
|pi*n       pi*n           2  2                                    
<                        pi *n                                     
|                                                                  
|               0                             otherwise            
\
{−4cos⁡(πn)πn+1πn+3sin⁡(πn)π2n2for n>−∞∧n<∞∧n≠00otherwise\begin{cases} - \frac{4 \cos{\left(\pi n \right)}}{\pi n} + \frac{1}{\pi n} + \frac{3 \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     4*cos(pi*n)   3*sin(pi*n)                                  
|---- - ----------- + -----------  for And(n > -oo, n < oo, n != 0)
|pi*n       pi*n           2  2                                    
<                        pi *n                                     
|                                                                  
|               0                             otherwise            
\
{−4cos⁡(πn)πn+1πn+3sin⁡(πn)π2n2for n>−∞∧n<∞∧n≠00otherwise\begin{cases} - \frac{4 \cos{\left(\pi n \right)}}{\pi n} + \frac{1}{\pi n} + \frac{3 \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) - 4*cos(pi*n)/(pi*n) + 3*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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