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Resolución de integrales/ x^3-x/ sin(pi*k*x)/ (x^3-x)*sin(pi*k*x)

Integral de (x^3-x)*sin(pi*k*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                        
  /                        
 |                         
 |  / 3    \               
 |  \x  - x/*sin(pi*k*x) dx
 |                         
/                          
-1
∫−11(x3−x)sin⁡(xπk) dx\int\limits_{-1}^{1} \left(x^{3} - x\right) \sin{\left(x \pi k \right)}\, dx 
Integral((x^3 - x)*sin((pi*k)*x), (x, -1, 1))
Respuesta (Indefinida) [src] 
                                   //                                 0                                   for k = 0\                                                                                                                 
                                   ||                                                                              |                                                                                                                 
                                   || //                   2                                          \            |                                                                                                                 
                                   || ||  2*sin(pi*k*x)   x *sin(pi*k*x)   2*x*cos(pi*k*x)            |            |                                                                    //               0                 for k = 0\
  /                                || ||- ------------- + -------------- + ---------------  for k != 0|            |                                                                    ||                                          |
 |                                 || ||        3  3           pi*k               2  2                |            |      //      0        for k = 0\     //      0        for k = 0\   || //sin(pi*k*x)               \            |
 | / 3    \                        || ||      pi *k                             pi *k                 |            |    3 ||                        |     ||                        |   || ||-----------  for pi*k != 0|            |
 | \x  - x/*sin(pi*k*x) dx = C - 3*|<-|<                                                              |            | + x *|<-cos(pi*k*x)            | - x*|<-cos(pi*k*x)            | + |<-|<    pi*k                  |            |
 |                                 || ||                         3                                    |            |      ||-------------  otherwise|     ||-------------  otherwise|   || ||                          |            |
/                                  || ||                        x                                     |            |      \\     pi*k               /     \\     pi*k               /   || \\     x         otherwise  /            |
                                   || ||                        --                          otherwise |            |                                                                    ||-------------------------------  otherwise|
                                   || ||                        3                                     |            |                                                                    \\              pi*k                        /
                                   || \\                                                              /            |                                                                                                                 
                                   ||-------------------------------------------------------------------  otherwise|                                                                                                                 
                                   \\                                pi*k                                          /
∫(x3−x)sin⁡(xπk) dx=C+x3({0for k=0−cos⁡(πkx)πkotherwise)−x({0for k=0−cos⁡(πkx)πkotherwise)+{0for k=0−{sin⁡(πkx)πkfor πk≠0xotherwiseπkotherwise−3({0for k=0−{x2sin⁡(πkx)πk+2xcos⁡(πkx)π2k2−2sin⁡(πkx)π3k3for k≠0x33otherwiseπkotherwise)\int \left(x^{3} - x\right) \sin{\left(x \pi k \right)}\, dx = C + x^{3} \left(\begin{cases} 0 & \text{for}\: k = 0 \\- \frac{\cos{\left(\pi k x \right)}}{\pi k} & \text{otherwise} \end{cases}\right) - 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} - 3 \left(\begin{cases} 0 & \text{for}\: k = 0 \\- \frac{\begin{cases} \frac{x^{2} \sin{\left(\pi k x \right)}}{\pi k} + \frac{2 x \cos{\left(\pi k x \right)}}{\pi^{2} k^{2}} - \frac{2 \sin{\left(\pi k x \right)}}{\pi^{3} k^{3}} & \text{for}\: k \neq 0 \\\frac{x^{3}}{3} & \text{otherwise} \end{cases}}{\pi k} & \text{otherwise} \end{cases}\right)
Simplificar
Respuesta [src] 
/  12*sin(pi*k)   4*sin(pi*k)   12*cos(pi*k)                                  
|- ------------ + ----------- + ------------  for And(k > -oo, k < oo, k != 0)
|       4  4           2  2          3  3                                     
<     pi *k          pi *k         pi *k                                      
|                                                                             
|                     0                                  otherwise            
\
{4sin⁡(πk)π2k2+12cos⁡(πk)π3k3−12sin⁡(πk)π4k4for k>−∞∧k<∞∧k≠00otherwise\begin{cases} \frac{4 \sin{\left(\pi k \right)}}{\pi^{2} k^{2}} + \frac{12 \cos{\left(\pi k \right)}}{\pi^{3} k^{3}} - \frac{12 \sin{\left(\pi k \right)}}{\pi^{4} k^{4}} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq 0 \\0 & \text{otherwise} \end{cases} 
=
= 
/  12*sin(pi*k)   4*sin(pi*k)   12*cos(pi*k)                                  
|- ------------ + ----------- + ------------  for And(k > -oo, k < oo, k != 0)
|       4  4           2  2          3  3                                     
<     pi *k          pi *k         pi *k                                      
|                                                                             
|                     0                                  otherwise            
\
{4sin⁡(πk)π2k2+12cos⁡(πk)π3k3−12sin⁡(πk)π4k4for k>−∞∧k<∞∧k≠00otherwise\begin{cases} \frac{4 \sin{\left(\pi k \right)}}{\pi^{2} k^{2}} + \frac{12 \cos{\left(\pi k \right)}}{\pi^{3} k^{3}} - \frac{12 \sin{\left(\pi k \right)}}{\pi^{4} k^{4}} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq 0 \\0 & \text{otherwise} \end{cases} 
Piecewise((-12*sin(pi*k)/(pi^4*k^4) + 4*sin(pi*k)/(pi^2*k^2) + 12*cos(pi*k)/(pi^3*k^3), (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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