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Resolución de integrales/ i*n*x/ (3*x-3)*sin(2*pi*n*x/3)

Integral de (3*x-3)*sin(2*pi*n*x/3) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                           
  /                           
 |                            
 |               /2*pi*n*x\   
 |  (3*x - 3)*sin|--------| dx
 |               \   3    /   
 |                            
/                             
0
∫01(3x−3)sin⁡(x2πn3) dx\int\limits_{0}^{1} \left(3 x - 3\right) \sin{\left(\frac{x 2 \pi n}{3} \right)}\, dx 
Integral((3*x - 3)*sin((((2*pi)*n)*x)/3), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                      //                  0                     for n = 0\                                                                        
                                      ||                                                 |                                                                        
  /                                   ||   //     /2*pi*n*x\                 \           |     //       0          for n = 0\       //       0          for n = 0\
 |                                    ||   ||3*sin|--------|                 |           |     ||                           |       ||                           |
 |              /2*pi*n*x\            ||   ||     \   3    /      2*pi*n     |           |     ||      /2*pi*n*x\           |       ||      /2*pi*n*x\           |
 | (3*x - 3)*sin|--------| dx = C - 3*|<-3*|<---------------  for ------ != 0|           | - 3*|<-3*cos|--------|           | + 3*x*|<-3*cos|--------|           |
 |              \   3    /            ||   ||     2*pi*n            3        |           |     ||      \   3    /           |       ||      \   3    /           |
 |                                    ||   ||                                |           |     ||----------------  otherwise|       ||----------------  otherwise|
/                                     ||   \\       x            otherwise   /           |     \\     2*pi*n                /       \\     2*pi*n                /
                                      ||--------------------------------------  otherwise|                                                                        
                                      \\                2*pi*n                           /
∫(3x−3)sin⁡(x2πn3) dx=C+3x({0for n=0−3cos⁡(2πnx3)2πnotherwise)−3({0for n=0−3({3sin⁡(2πnx3)2πnfor 2πn3≠0xotherwise)2πnotherwise)−3({0for n=0−3cos⁡(2πnx3)2πnotherwise)\int \left(3 x - 3\right) \sin{\left(\frac{x 2 \pi n}{3} \right)}\, dx = C + 3 x \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{3 \cos{\left(\frac{2 \pi n x}{3} \right)}}{2 \pi n} & \text{otherwise} \end{cases}\right) - 3 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{3 \left(\begin{cases} \frac{3 \sin{\left(\frac{2 \pi n x}{3} \right)}}{2 \pi n} & \text{for}\: \frac{2 \pi n}{3} \neq 0 \\x & \text{otherwise} \end{cases}\right)}{2 \pi n} & \text{otherwise} \end{cases}\right) - 3 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{3 \cos{\left(\frac{2 \pi n x}{3} \right)}}{2 \pi n} & \text{otherwise} \end{cases}\right)
Simplificar
Respuesta [src] 
/                 /2*pi*n\                                  
|           27*sin|------|                                  
|    9            \  3   /                                  
|- ------ + --------------  for And(n > -oo, n < oo, n != 0)
<  2*pi*n          2  2                                     
|              4*pi *n                                      
|                                                           
|            0                         otherwise            
\
{−92πn+27sin⁡(2πn3)4π2n2for n>−∞∧n<∞∧n≠00otherwise\begin{cases} - \frac{9}{2 \pi n} + \frac{27 \sin{\left(\frac{2 \pi n}{3} \right)}}{4 \pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases} 
=
= 
/                 /2*pi*n\                                  
|           27*sin|------|                                  
|    9            \  3   /                                  
|- ------ + --------------  for And(n > -oo, n < oo, n != 0)
<  2*pi*n          2  2                                     
|              4*pi *n                                      
|                                                           
|            0                         otherwise            
\
{−92πn+27sin⁡(2πn3)4π2n2for n>−∞∧n<∞∧n≠00otherwise\begin{cases} - \frac{9}{2 \pi n} + \frac{27 \sin{\left(\frac{2 \pi n}{3} \right)}}{4 \pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases} 
Piecewise((-9/(2*pi*n) + 27*sin(2*pi*n/3)/(4*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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