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Resolución de integrales/ sinnx/ (x-π)sinnxdx

Integral de (x-π)sinnxdx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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