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Resolución de integrales/ sin(kx)/ (4-2*x)*sin(kx)

Integral de (4-2*x)*sin(kx) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi                      
  /                      
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 |  (4 - 2*x)*sin(k*x) dx
 |                       
/                        
0
∫0π(4−2x)sin⁡(kx) dx\int\limits_{0}^{\pi} \left(4 - 2 x\right) \sin{\left(k x \right)}\, dx 
Integral((4 - 2*x)*sin(k*x), (x, 0, pi))
Respuesta (Indefinida) [src] 
                                 //            0              for k = 0\                                                            
                                 ||                                    |                                                            
  /                              || //sin(k*x)            \            |     //    0       for k = 0\       //    0       for k = 0\
 |                               || ||--------  for k != 0|            |     ||                     |       ||                     |
 | (4 - 2*x)*sin(k*x) dx = C + 2*|<-|<   k                |            | + 4*|<-cos(k*x)            | - 2*x*|<-cos(k*x)            |
 |                               || ||                    |            |     ||----------  otherwise|       ||----------  otherwise|
/                                || \\   x      otherwise /            |     \\    k                /       \\    k                /
                                 ||-------------------------  otherwise|                                                            
                                 \\            k                       /
∫(4−2x)sin⁡(kx) dx=C−2x({0for k=0−cos⁡(kx)kotherwise)+2({0for k=0−{sin⁡(kx)kfor k≠0xotherwisekotherwise)+4({0for k=0−cos⁡(kx)kotherwise)\int \left(4 - 2 x\right) \sin{\left(k x \right)}\, dx = C - 2 x \left(\begin{cases} 0 & \text{for}\: k = 0 \\- \frac{\cos{\left(k x \right)}}{k} & \text{otherwise} \end{cases}\right) + 2 \left(\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}\right) + 4 \left(\begin{cases} 0 & \text{for}\: k = 0 \\- \frac{\cos{\left(k x \right)}}{k} & \text{otherwise} \end{cases}\right)
Simplificar
Respuesta [src] 
/4   4*cos(pi*k)   2*sin(pi*k)   2*pi*cos(pi*k)                                  
|- - ----------- - ----------- + --------------  for And(k > -oo, k < oo, k != 0)
|k        k              2             k                                         
<                       k                                                        
|                                                                                
|                      0                                    otherwise            
\
{−4cos⁡(πk)k+2πcos⁡(πk)k+4k−2sin⁡(πk)k2for k>−∞∧k<∞∧k≠00otherwise\begin{cases} - \frac{4 \cos{\left(\pi k \right)}}{k} + \frac{2 \pi \cos{\left(\pi k \right)}}{k} + \frac{4}{k} - \frac{2 \sin{\left(\pi k \right)}}{k^{2}} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq 0 \\0 & \text{otherwise} \end{cases} 
=
= 
/4   4*cos(pi*k)   2*sin(pi*k)   2*pi*cos(pi*k)                                  
|- - ----------- - ----------- + --------------  for And(k > -oo, k < oo, k != 0)
|k        k              2             k                                         
<                       k                                                        
|                                                                                
|                      0                                    otherwise            
\
{−4cos⁡(πk)k+2πcos⁡(πk)k+4k−2sin⁡(πk)k2for k>−∞∧k<∞∧k≠00otherwise\begin{cases} - \frac{4 \cos{\left(\pi k \right)}}{k} + \frac{2 \pi \cos{\left(\pi k \right)}}{k} + \frac{4}{k} - \frac{2 \sin{\left(\pi k \right)}}{k^{2}} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq 0 \\0 & \text{otherwise} \end{cases} 
Piecewise((4/k - 4*cos(pi*k)/k - 2*sin(pi*k)/k^2 + 2*pi*cos(pi*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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