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Resolución de integrales/ x^2-x/ (x^2-x)*cos(x*K)

Integral de (x^2-x)*cos(x*K) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi                     
  /                     
 |                      
 |  / 2    \            
 |  \x  - x/*cos(x*k) dx
 |                      
/                       
-pi
∫−ππ(x2−x)cos⁡(kx) dx\int\limits_{- \pi}^{\pi} \left(x^{2} - x\right) \cos{\left(k x \right)}\, dx 
Integral((x^2 - x)*cos(x*k), (x, -pi, pi))
Respuesta (Indefinida) [src] 
                                //                 3                           \                                                                                               
                                ||                x                            |                                                          //           2                      \
                                ||                --                  for k = 0|                                                          ||          x                       |
                                ||                3                            |                                                          ||          --             for k = 0|
  /                             ||                                             |                                                          ||          2                       |
 |                              ||/sin(k*x)   x*cos(k*x)                       |      //   x      for k = 0\     //   x      for k = 0\   ||                                  |
 | / 2    \                     |||-------- - ----------  for k != 0           |    2 ||                   |     ||                   |   ||/-cos(k*x)                        |
 | \x  - x/*cos(x*k) dx = C - 2*|<|    2          k                            | + x *|
∫(x2−x)cos⁡(kx) dx=C+x2({xfor k=0sin⁡(kx)kotherwise)−x({xfor k=0sin⁡(kx)kotherwise)+{x22for k=0{−cos⁡(kx)kfor k≠00otherwisekotherwise−2({x33for k=0{−xcos⁡(kx)k+sin⁡(kx)k2for k≠00otherwisekotherwise)\int \left(x^{2} - x\right) \cos{\left(k x \right)}\, dx = C + x^{2} \left(\begin{cases} x & \text{for}\: k = 0 \\\frac{\sin{\left(k x \right)}}{k} & \text{otherwise} \end{cases}\right) - x \left(\begin{cases} x & \text{for}\: k = 0 \\\frac{\sin{\left(k x \right)}}{k} & \text{otherwise} \end{cases}\right) + \begin{cases} \frac{x^{2}}{2} & \text{for}\: k = 0 \\\frac{\begin{cases} - \frac{\cos{\left(k x \right)}}{k} & \text{for}\: k \neq 0 \\0 & \text{otherwise} \end{cases}}{k} & \text{otherwise} \end{cases} - 2 \left(\begin{cases} \frac{x^{3}}{3} & \text{for}\: k = 0 \\\frac{\begin{cases} - \frac{x \cos{\left(k x \right)}}{k} + \frac{\sin{\left(k x \right)}}{k^{2}} & \text{for}\: k \neq 0 \\0 & \text{otherwise} \end{cases}}{k} & \text{otherwise} \end{cases}\right)
Simplificar
Respuesta [src] 
/                    2                                                             
|  4*sin(pi*k)   2*pi *sin(pi*k)   4*pi*cos(pi*k)                                  
|- ----------- + --------------- + --------------  for And(k > -oo, k < oo, k != 0)
|        3              k                 2                                        
|       k                                k                                         
<                                                                                  
|                         3                                                        
|                     2*pi                                                         
|                     -----                                   otherwise            
|                       3                                                          
\
{2π2sin⁡(πk)k+4πcos⁡(πk)k2−4sin⁡(πk)k3for k>−∞∧k<∞∧k≠02π33otherwise\begin{cases} \frac{2 \pi^{2} \sin{\left(\pi k \right)}}{k} + \frac{4 \pi \cos{\left(\pi k \right)}}{k^{2}} - \frac{4 \sin{\left(\pi k \right)}}{k^{3}} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq 0 \\\frac{2 \pi^{3}}{3} & \text{otherwise} \end{cases} 
=
= 
/                    2                                                             
|  4*sin(pi*k)   2*pi *sin(pi*k)   4*pi*cos(pi*k)                                  
|- ----------- + --------------- + --------------  for And(k > -oo, k < oo, k != 0)
|        3              k                 2                                        
|       k                                k                                         
<                                                                                  
|                         3                                                        
|                     2*pi                                                         
|                     -----                                   otherwise            
|                       3                                                          
\
{2π2sin⁡(πk)k+4πcos⁡(πk)k2−4sin⁡(πk)k3for k>−∞∧k<∞∧k≠02π33otherwise\begin{cases} \frac{2 \pi^{2} \sin{\left(\pi k \right)}}{k} + \frac{4 \pi \cos{\left(\pi k \right)}}{k^{2}} - \frac{4 \sin{\left(\pi k \right)}}{k^{3}} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq 0 \\\frac{2 \pi^{3}}{3} & \text{otherwise} \end{cases} 
Piecewise((-4*sin(pi*k)/k^3 + 2*pi^2*sin(pi*k)/k + 4*pi*cos(pi*k)/k^2, (k > -oo)∧(k < oo)∧(Ne(k, 0))), (2*pi^3/3, True))

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

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