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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  3                         
  /                         
 |                          
 |  2*(3 - x)    /pi*n*x\   
 |  ---------*sin|------| dx
 |      3        \  3   /   
 |                          
/                           
1
∫132(3−x)3sin⁡(xπn3) dx\int\limits_{1}^{3} \frac{2 \left(3 - x\right)}{3} \sin{\left(\frac{x \pi n}{3} \right)}\, dx 
Integral((2*(3 - x)/3)*sin(((pi*n)*x)/3), (x, 1, 3))
Respuesta (Indefinida) [src] 
                                                                     //                0                   for n = 0\                                   
                                                                     ||                                             |                                   
                                                                     ||   //     /pi*n*x\               \           |                                   
                                                                     ||   ||3*sin|------|               |           |                                   
                                                                     ||   ||     \  3   /      pi*n     |           |       //      0         for n = 0\
                                                                   2*|<-3*|<-------------  for ---- != 0|           |       ||                         |
                                                                     ||   ||     pi*n           3       |           |       ||      /pi*n*x\           |
                                                                     ||   ||                            |           |   2*x*|<-3*cos|------|           |
  /                                 //      0         for n = 0\     ||   \\      x          otherwise  /           |       ||      \  3   /           |
 |                                  ||                         |     ||----------------------------------  otherwise|       ||--------------  otherwise|
 | 2*(3 - x)    /pi*n*x\            ||      /pi*n*x\           |     \\               pi*n                          /       \\     pi*n                /
 | ---------*sin|------| dx = C + 2*|<-3*cos|------|           | + -------------------------------------------------- - --------------------------------
 |     3        \  3   /            ||      \  3   /           |                           3                                           3                
 |                                  ||--------------  otherwise|                                                                                        
/                                   \\     pi*n                /
∫2(3−x)3sin⁡(xπn3) dx=C−2x({0for n=0−3cos⁡(πnx3)πnotherwise)3+2({0for n=0−3({3sin⁡(πnx3)πnfor πn3≠0xotherwise)πnotherwise)3+2({0for n=0−3cos⁡(πnx3)πnotherwise)\int \frac{2 \left(3 - x\right)}{3} \sin{\left(\frac{x \pi n}{3} \right)}\, dx = C - \frac{2 x \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{3 \cos{\left(\frac{\pi n x}{3} \right)}}{\pi n} & \text{otherwise} \end{cases}\right)}{3} + \frac{2 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{3 \left(\begin{cases} \frac{3 \sin{\left(\frac{\pi n x}{3} \right)}}{\pi n} & \text{for}\: \frac{\pi n}{3} \neq 0 \\x & \text{otherwise} \end{cases}\right)}{\pi n} & \text{otherwise} \end{cases}\right)}{3} + 2 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{3 \cos{\left(\frac{\pi n x}{3} \right)}}{\pi n} & \text{otherwise} \end{cases}\right)
Simplificar
Respuesta [src] 
/                     /pi*n\        /pi*n\                                  
|                4*cos|----|   6*sin|----|                                  
|  6*sin(pi*n)        \ 3  /        \ 3  /                                  
|- ----------- + ----------- + -----------  for And(n > -oo, n < oo, n != 0)
<       2  2         pi*n           2  2                                    
|     pi *n                       pi *n                                     
|                                                                           
|                    0                                 otherwise            
\
{4cos⁡(πn3)πn+6sin⁡(πn3)π2n2−6sin⁡(πn)π2n2for n>−∞∧n<∞∧n≠00otherwise\begin{cases} \frac{4 \cos{\left(\frac{\pi n}{3} \right)}}{\pi n} + \frac{6 \sin{\left(\frac{\pi n}{3} \right)}}{\pi^{2} n^{2}} - \frac{6 \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} 
=
= 
/                     /pi*n\        /pi*n\                                  
|                4*cos|----|   6*sin|----|                                  
|  6*sin(pi*n)        \ 3  /        \ 3  /                                  
|- ----------- + ----------- + -----------  for And(n > -oo, n < oo, n != 0)
<       2  2         pi*n           2  2                                    
|     pi *n                       pi *n                                     
|                                                                           
|                    0                                 otherwise            
\
{4cos⁡(πn3)πn+6sin⁡(πn3)π2n2−6sin⁡(πn)π2n2for n>−∞∧n<∞∧n≠00otherwise\begin{cases} \frac{4 \cos{\left(\frac{\pi n}{3} \right)}}{\pi n} + \frac{6 \sin{\left(\frac{\pi n}{3} \right)}}{\pi^{2} n^{2}} - \frac{6 \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((-6*sin(pi*n)/(pi^2*n^2) + 4*cos(pi*n/3)/(pi*n) + 6*sin(pi*n/3)/(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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