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Resolución de integrales/ (4-x)/ (4-x)*cos(pi*x*K/2)

Integral de (4-x)*cos(pi*x*K/2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  4                       
  /                       
 |                        
 |             /pi*x*k\   
 |  (4 - x)*cos|------| dx
 |             \  2   /   
 |                        
/                         
0
∫04(4−x)cos⁡(kπx2) dx\int\limits_{0}^{4} \left(4 - x\right) \cos{\left(\frac{k \pi x}{2} \right)}\, dx 
Integral((4 - x)*cos(((pi*x)*k)/2), (x, 0, 4))
Respuesta (Indefinida) [src] 
                                                                                                //                 2                           \
                                                                                                ||                x                            |
                                                                                                ||                --                  for k = 0|
                                                                                                ||                2                            |
  /                               //      x        for k = 0\     //      x        for k = 0\   ||                                             |
 |                                ||                        |     ||                        |   ||  //      /pi*k*x\               \           |
 |            /pi*x*k\            ||     /pi*k*x\           |     ||     /pi*k*x\           |   ||  ||-2*cos|------|               |           |
 | (4 - x)*cos|------| dx = C + 4*|<2*sin|------|           | - x*|<2*sin|------|           | + |<  ||      \  2   /      pi*k     |           |
 |            \  2   /            ||     \  2   /           |     ||     \  2   /           |   ||2*|<--------------  for ---- != 0|           |
 |                                ||-------------  otherwise|     ||-------------  otherwise|   ||  ||     pi*k            2       |           |
/                                 \\     pi*k               /     \\     pi*k               /   ||  ||                             |           |
                                                                                                ||  \\      0           otherwise  /           |
                                                                                                ||----------------------------------  otherwise|
                                                                                                ||               pi*k                          |
                                                                                                \\                                             /
∫(4−x)cos⁡(kπx2) dx=C−x({xfor k=02sin⁡(πkx2)πkotherwise)+4({xfor k=02sin⁡(πkx2)πkotherwise)+{x22for k=02({−2cos⁡(πkx2)πkfor πk2≠00otherwise)πkotherwise\int \left(4 - x\right) \cos{\left(\frac{k \pi x}{2} \right)}\, dx = C - x \left(\begin{cases} x & \text{for}\: k = 0 \\\frac{2 \sin{\left(\frac{\pi k x}{2} \right)}}{\pi k} & \text{otherwise} \end{cases}\right) + 4 \left(\begin{cases} x & \text{for}\: k = 0 \\\frac{2 \sin{\left(\frac{\pi k x}{2} \right)}}{\pi k} & \text{otherwise} \end{cases}\right) + \begin{cases} \frac{x^{2}}{2} & \text{for}\: k = 0 \\\frac{2 \left(\begin{cases} - \frac{2 \cos{\left(\frac{\pi k x}{2} \right)}}{\pi k} & \text{for}\: \frac{\pi k}{2} \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{\pi k} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/  4      4*cos(2*pi*k)                                  
|------ - -------------  for And(k > -oo, k < oo, k != 0)
|  2  2         2  2
{−4cos⁡(2πk)π2k2+4π2k2for k>−∞∧k<∞∧k≠08otherwise\begin{cases} - \frac{4 \cos{\left(2 \pi k \right)}}{\pi^{2} k^{2}} + \frac{4}{\pi^{2} k^{2}} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq 0 \\8 & \text{otherwise} \end{cases} 
=
= 
/  4      4*cos(2*pi*k)                                  
|------ - -------------  for And(k > -oo, k < oo, k != 0)
|  2  2         2  2
{−4cos⁡(2πk)π2k2+4π2k2for k>−∞∧k<∞∧k≠08otherwise\begin{cases} - \frac{4 \cos{\left(2 \pi k \right)}}{\pi^{2} k^{2}} + \frac{4}{\pi^{2} k^{2}} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq 0 \\8 & \text{otherwise} \end{cases} 
Piecewise((4/(pi^2*k^2) - 4*cos(2*pi*k)/(pi^2*k^2), (k > -oo)∧(k < oo)∧(Ne(k, 0))), (8, True))

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

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