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Resolución de integrales/ sin(nx)/ sign(x)/ (x+sign(x))sin(nx)

Integral de (x+sign(x))sin(nx) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi                          
  /                          
 |                           
 |  (x + sign(x))*sin(n*x) dx
 |                           
/                            
0
∫0π(x+sign⁡(x))sin⁡(nx) dx\int\limits_{0}^{\pi} \left(x + \operatorname{sign}{\left(x \right)}\right) \sin{\left(n x \right)}\, dx 
Integral((x + sign(x))*sin(n*x), (x, 0, pi))
Respuesta (Indefinida) [src] 
                                   //            0              for n = 0\                                                      
                                   ||                                    |                                                      
  /                                || //sin(n*x)            \            |     //    0       for n = 0\     /                   
 |                                 || ||--------  for n != 0|            |     ||                     |    |                    
 | (x + sign(x))*sin(n*x) dx = C - |<-|<   n                |            | + x*|<-cos(n*x)            | +  | sign(x)*sin(n*x) dx
 |                                 || ||                    |            |     ||----------  otherwise|    |                    
/                                  || \\   x      otherwise /            |     \\    n                /   /                     
                                   ||-------------------------  otherwise|                                                      
                                   \\            n                       /
∫(x+sign⁡(x))sin⁡(nx) dx=C+x({0for n=0−cos⁡(nx)notherwise)−{0for n=0−{sin⁡(nx)nfor n≠0xotherwisenotherwise+∫sin⁡(nx)sign⁡(x) dx\int \left(x + \operatorname{sign}{\left(x \right)}\right) \sin{\left(n x \right)}\, dx = C + x \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\cos{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right) - \begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\begin{cases} \frac{\sin{\left(n x \right)}}{n} & \text{for}\: n \neq 0 \\x & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases} + \int \sin{\left(n x \right)} \operatorname{sign}{\left(x \right)}\, dx
Simplificar
Respuesta [src] 
/1   sin(pi*n)   cos(pi*n)   pi*cos(pi*n)                                  
|- + --------- - --------- - ------------  for And(n > -oo, n < oo, n != 0)
|n        2          n            n                                        
<        n                                                                 
|                                                                          
|                   0                                 otherwise            
\
{−πcos⁡(πn)n−cos⁡(πn)n+1n+sin⁡(πn)n2for n>−∞∧n<∞∧n≠00otherwise\begin{cases} - \frac{\pi \cos{\left(\pi n \right)}}{n} - \frac{\cos{\left(\pi n \right)}}{n} + \frac{1}{n} + \frac{\sin{\left(\pi n \right)}}{n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases} 
=
= 
/1   sin(pi*n)   cos(pi*n)   pi*cos(pi*n)                                  
|- + --------- - --------- - ------------  for And(n > -oo, n < oo, n != 0)
|n        2          n            n                                        
<        n                                                                 
|                                                                          
|                   0                                 otherwise            
\
{−πcos⁡(πn)n−cos⁡(πn)n+1n+sin⁡(πn)n2for n>−∞∧n<∞∧n≠00otherwise\begin{cases} - \frac{\pi \cos{\left(\pi n \right)}}{n} - \frac{\cos{\left(\pi n \right)}}{n} + \frac{1}{n} + \frac{\sin{\left(\pi n \right)}}{n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases} 
Piecewise((1/n + sin(pi*n)/n^2 - cos(pi*n)/n - pi*cos(pi*n)/n, (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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