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Resolución de integrales/ sin(kx)/ xsin(kx)

Integral de xsin(kx) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1              
  /              
 |               
 |  x*sin(k*x) dx
 |               
/                
0
∫01xsin⁡(kx) dx\int\limits_{0}^{1} x \sin{\left(k x \right)}\, dx 
Integral(x*sin(k*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
                       //            0              for k = 0\                             
                       ||                                    |                             
  /                    || //sin(k*x)            \            |     //    0       for k = 0\
 |                     || ||--------  for k != 0|            |     ||                     |
 | x*sin(k*x) dx = C - |<-|<   k                |            | + x*|<-cos(k*x)            |
 |                     || ||                    |            |     ||----------  otherwise|
/                      || \\   x      otherwise /            |     \\    k                /
                       ||-------------------------  otherwise|                             
                       \\            k                       /
∫xsin⁡(kx) dx=C+x({0for k=0−cos⁡(kx)kotherwise)−{0for k=0−{sin⁡(kx)kfor k≠0xotherwisekotherwise\int x \sin{\left(k x \right)}\, dx = C + x \left(\begin{cases} 0 & \text{for}\: k = 0 \\- \frac{\cos{\left(k x \right)}}{k} & \text{otherwise} \end{cases}\right) - \begin{cases} 0 & \text{for}\: k = 0 \\- \frac{\begin{cases} \frac{\sin{\left(k x \right)}}{k} & \text{for}\: k \neq 0 \\x & \text{otherwise} \end{cases}}{k} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/sin(k)   cos(k)                                  
|------ - ------  for And(k > -oo, k < oo, k != 0)
|   2       k                                     
<  k                                              
|                                                 
|       0                    otherwise            
\
{−cos⁡(k)k+sin⁡(k)k2for k>−∞∧k<∞∧k≠00otherwise\begin{cases} - \frac{\cos{\left(k \right)}}{k} + \frac{\sin{\left(k \right)}}{k^{2}} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq 0 \\0 & \text{otherwise} \end{cases} 
=
= 
/sin(k)   cos(k)                                  
|------ - ------  for And(k > -oo, k < oo, k != 0)
|   2       k                                     
<  k                                              
|                                                 
|       0                    otherwise            
\
{−cos⁡(k)k+sin⁡(k)k2for k>−∞∧k<∞∧k≠00otherwise\begin{cases} - \frac{\cos{\left(k \right)}}{k} + \frac{\sin{\left(k \right)}}{k^{2}} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq 0 \\0 & \text{otherwise} \end{cases} 
Piecewise((sin(k)/k^2 - cos(k)/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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