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Resolución de integrales/ i*n*x/ (x+4)/ (x+4)*sin(pi*n*x/4)

Integral de (x+4)*sin(pi*n*x/4) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  4                       
  /                       
 |                        
 |             /pi*n*x\   
 |  (x + 4)*sin|------| dx
 |             \  4   /   
 |                        
/                         
0
∫04(x+4)sin⁡(xπn4) dx\int\limits_{0}^{4} \left(x + 4\right) \sin{\left(\frac{x \pi n}{4} \right)}\, dx 
Integral((x + 4)*sin(((pi*n)*x)/4), (x, 0, 4))
Respuesta (Indefinida) [src] 
                                //                0                   for n = 0\                                                                  
                                ||                                             |                                                                  
  /                             ||   //     /pi*n*x\               \           |     //      0         for n = 0\     //      0         for n = 0\
 |                              ||   ||4*sin|------|               |           |     ||                         |     ||                         |
 |            /pi*n*x\          ||   ||     \  4   /      pi*n     |           |     ||      /pi*n*x\           |     ||      /pi*n*x\           |
 | (x + 4)*sin|------| dx = C - |<-4*|<-------------  for ---- != 0|           | + 4*|<-4*cos|------|           | + x*|<-4*cos|------|           |
 |            \  4   /          ||   ||     pi*n           4       |           |     ||      \  4   /           |     ||      \  4   /           |
 |                              ||   ||                            |           |     ||--------------  otherwise|     ||--------------  otherwise|
/                               ||   \\      x          otherwise  /           |     \\     pi*n                /     \\     pi*n                /
                                ||----------------------------------  otherwise|                                                                  
                                \\               pi*n                          /
∫(x+4)sin⁡(xπn4) dx=C+x({0for n=0−4cos⁡(πnx4)πnotherwise)−{0for n=0−4({4sin⁡(πnx4)πnfor πn4≠0xotherwise)πnotherwise+4({0for n=0−4cos⁡(πnx4)πnotherwise)\int \left(x + 4\right) \sin{\left(\frac{x \pi n}{4} \right)}\, dx = C + x \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{4 \cos{\left(\frac{\pi n x}{4} \right)}}{\pi n} & \text{otherwise} \end{cases}\right) - \begin{cases} 0 & \text{for}\: n = 0 \\- \frac{4 \left(\begin{cases} \frac{4 \sin{\left(\frac{\pi n x}{4} \right)}}{\pi n} & \text{for}\: \frac{\pi n}{4} \neq 0 \\x & \text{otherwise} \end{cases}\right)}{\pi n} & \text{otherwise} \end{cases} + 4 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{4 \cos{\left(\frac{\pi n x}{4} \right)}}{\pi n} & \text{otherwise} \end{cases}\right)
Simplificar
Respuesta [src] 
/ 16    32*cos(pi*n)   16*sin(pi*n)                                  
|---- - ------------ + ------------  for And(n > -oo, n < oo, n != 0)
|pi*n       pi*n            2  2                                     
<                         pi *n                                      
|                                                                    
|                0                              otherwise            
\
{−32cos⁡(πn)πn+16πn+16sin⁡(πn)π2n2for n>−∞∧n<∞∧n≠00otherwise\begin{cases} - \frac{32 \cos{\left(\pi n \right)}}{\pi n} + \frac{16}{\pi n} + \frac{16 \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} 
=
= 
/ 16    32*cos(pi*n)   16*sin(pi*n)                                  
|---- - ------------ + ------------  for And(n > -oo, n < oo, n != 0)
|pi*n       pi*n            2  2                                     
<                         pi *n                                      
|                                                                    
|                0                              otherwise            
\
{−32cos⁡(πn)πn+16πn+16sin⁡(πn)π2n2for n>−∞∧n<∞∧n≠00otherwise\begin{cases} - \frac{32 \cos{\left(\pi n \right)}}{\pi n} + \frac{16}{\pi n} + \frac{16 \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((16/(pi*n) - 32*cos(pi*n)/(pi*n) + 16*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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