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Resolución de integrales/ (6-x)/ (6-x)*cos(n*x*pi/4)

Integral de (6-x)*cos(n*x*pi/4) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  4                       
  /                       
 |                        
 |             /n*x*pi\   
 |  (6 - x)*cos|------| dx
 |             \  4   /   
 |                        
/                         
3
∫34(6−x)cos⁡(πnx4) dx\int\limits_{3}^{4} \left(6 - x\right) \cos{\left(\frac{\pi n x}{4} \right)}\, dx 
Integral((6 - x)*cos(((n*x)*pi)/4), (x, 3, 4))
Respuesta (Indefinida) [src] 
                                                                                                //                 2                           \
                                                                                                ||                x                            |
                                                                                                ||                --                  for n = 0|
                                                                                                ||                2                            |
  /                               //      x        for n = 0\     //      x        for n = 0\   ||                                             |
 |                                ||                        |     ||                        |   ||  //      /pi*n*x\               \           |
 |            /n*x*pi\            ||     /pi*n*x\           |     ||     /pi*n*x\           |   ||  ||-4*cos|------|               |           |
 | (6 - x)*cos|------| dx = C + 6*|<4*sin|------|           | - x*|<4*sin|------|           | + |<  ||      \  4   /      pi*n     |           |
 |            \  4   /            ||     \  4   /           |     ||     \  4   /           |   ||4*|<--------------  for ---- != 0|           |
 |                                ||-------------  otherwise|     ||-------------  otherwise|   ||  ||     pi*n            4       |           |
/                                 \\     pi*n               /     \\     pi*n               /   ||  ||                             |           |
                                                                                                ||  \\      0           otherwise  /           |
                                                                                                ||----------------------------------  otherwise|
                                                                                                ||               pi*n                          |
                                                                                                \\                                             /
∫(6−x)cos⁡(πnx4) dx=C−x({xfor n=04sin⁡(πnx4)πnotherwise)+6({xfor n=04sin⁡(πnx4)πnotherwise)+{x22for n=04({−4cos⁡(πnx4)πnfor πn4≠00otherwise)πnotherwise\int \left(6 - x\right) \cos{\left(\frac{\pi n x}{4} \right)}\, dx = C - x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{4 \sin{\left(\frac{\pi n x}{4} \right)}}{\pi n} & \text{otherwise} \end{cases}\right) + 6 \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{4 \sin{\left(\frac{\pi n x}{4} \right)}}{\pi n} & \text{otherwise} \end{cases}\right) + \begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{4 \left(\begin{cases} - \frac{4 \cos{\left(\frac{\pi n x}{4} \right)}}{\pi n} & \text{for}\: \frac{\pi n}{4} \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{\pi n} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/                       /3*pi*n\                       /3*pi*n\                                  
|                 12*sin|------|                 16*cos|------|                                  
|  16*cos(pi*n)         \  4   /   8*sin(pi*n)         \  4   /                                  
|- ------------ - -------------- + ----------- + --------------  for And(n > -oo, n < oo, n != 0)
<       2  2           pi*n            pi*n            2  2                                      
|     pi *n                                          pi *n                                       
|                                                                                                
|                             5/2                                           otherwise            
\
{−12sin⁡(3πn4)πn+8sin⁡(πn)πn+16cos⁡(3πn4)π2n2−16cos⁡(πn)π2n2for n>−∞∧n<∞∧n≠052otherwise\begin{cases} - \frac{12 \sin{\left(\frac{3 \pi n}{4} \right)}}{\pi n} + \frac{8 \sin{\left(\pi n \right)}}{\pi n} + \frac{16 \cos{\left(\frac{3 \pi n}{4} \right)}}{\pi^{2} n^{2}} - \frac{16 \cos{\left(\pi n \right)}}{\pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{5}{2} & \text{otherwise} \end{cases} 
=
= 
/                       /3*pi*n\                       /3*pi*n\                                  
|                 12*sin|------|                 16*cos|------|                                  
|  16*cos(pi*n)         \  4   /   8*sin(pi*n)         \  4   /                                  
|- ------------ - -------------- + ----------- + --------------  for And(n > -oo, n < oo, n != 0)
<       2  2           pi*n            pi*n            2  2                                      
|     pi *n                                          pi *n                                       
|                                                                                                
|                             5/2                                           otherwise            
\
{−12sin⁡(3πn4)πn+8sin⁡(πn)πn+16cos⁡(3πn4)π2n2−16cos⁡(πn)π2n2for n>−∞∧n<∞∧n≠052otherwise\begin{cases} - \frac{12 \sin{\left(\frac{3 \pi n}{4} \right)}}{\pi n} + \frac{8 \sin{\left(\pi n \right)}}{\pi n} + \frac{16 \cos{\left(\frac{3 \pi n}{4} \right)}}{\pi^{2} n^{2}} - \frac{16 \cos{\left(\pi n \right)}}{\pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{5}{2} & \text{otherwise} \end{cases} 
Piecewise((-16*cos(pi*n)/(pi^2*n^2) - 12*sin(3*pi*n/4)/(pi*n) + 8*sin(pi*n)/(pi*n) + 16*cos(3*pi*n/4)/(pi^2*n^2), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (5/2, True))

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

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