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Resolución de integrales/ sin(pi*k*x)/ xsin(pi*k*x)

Integral de xsin(pi*k*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 1/2                
  /                 
 |                  
 |  x*sin(pi*k*x) dx
 |                  
/                   
0
$$\int\limits_{0}^{\frac{1}{2}} x \sin{\left(x \pi k \right)}\, dx$$ 
Integral(x*sin((pi*k)*x), (x, 0, 1/2))
Respuesta (Indefinida) [src] 
                          //               0                 for k = 0\                                
                          ||                                          |                                
  /                       || //sin(pi*k*x)               \            |     //      0        for k = 0\
 |                        || ||-----------  for pi*k != 0|            |     ||                        |
 | x*sin(pi*k*x) dx = C - |<-|<    pi*k                  |            | + x*|<-cos(pi*k*x)            |
 |                        || ||                          |            |     ||-------------  otherwise|
/                         || \\     x         otherwise  /            |     \\     pi*k               /
                          ||-------------------------------  otherwise|                                
                          \\              pi*k                        /
$$\int x \sin{\left(x \pi k \right)}\, dx = C + x \left(\begin{cases} 0 & \text{for}\: k = 0 \\- \frac{\cos{\left(\pi k x \right)}}{\pi k} & \text{otherwise} \end{cases}\right) - \begin{cases} 0 & \text{for}\: k = 0 \\- \frac{\begin{cases} \frac{\sin{\left(\pi k x \right)}}{\pi k} & \text{for}\: \pi k \neq 0 \\x & \text{otherwise} \end{cases}}{\pi k} & \text{otherwise} \end{cases}$$
Simplificar
Respuesta [src] 
/   /pi*k\      /pi*k\                                  
|sin|----|   cos|----|                                  
|   \ 2  /      \ 2  /                                  
|--------- - ---------  for And(k > -oo, k < oo, k != 0)
<    2  2      2*pi*k                                   
|  pi *k                                                
|                                                       
|          0                       otherwise            
\
$$\begin{cases} - \frac{\cos{\left(\frac{\pi k}{2} \right)}}{2 \pi k} + \frac{\sin{\left(\frac{\pi k}{2} \right)}}{\pi^{2} k^{2}} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq 0 \\0 & \text{otherwise} \end{cases}$$ 
=
= 
/   /pi*k\      /pi*k\                                  
|sin|----|   cos|----|                                  
|   \ 2  /      \ 2  /                                  
|--------- - ---------  for And(k > -oo, k < oo, k != 0)
<    2  2      2*pi*k                                   
|  pi *k                                                
|                                                       
|          0                       otherwise            
\
$$\begin{cases} - \frac{\cos{\left(\frac{\pi k}{2} \right)}}{2 \pi k} + \frac{\sin{\left(\frac{\pi k}{2} \right)}}{\pi^{2} k^{2}} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq 0 \\0 & \text{otherwise} \end{cases}$$ 
Piecewise((sin(pi*k/2)/(pi^2*k^2) - cos(pi*k/2)/(2*pi*k), (k > -oo)∧(k < oo)∧(Ne(k, 0))), (0, True))

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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