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Resolución de integrales/ i*n*x/ (2x-1)/ (2x-1)*cos((pi*n*x)/2)

Integral de (2x-1)*cos((pi*n*x)/2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 1/2                        
  /                         
 |                          
 |               /pi*n*x\   
 |  (2*x - 1)*cos|------| dx
 |               \  2   /   
 |                          
/                           
0
∫012(2x−1)cos⁡(xπn2) dx\int\limits_{0}^{\frac{1}{2}} \left(2 x - 1\right) \cos{\left(\frac{x \pi n}{2} \right)}\, dx 
Integral((2*x - 1)*cos(((pi*n)*x)/2), (x, 0, 1/2))
Respuesta (Indefinida) [src] 
                                                                  //                 2                           \                                  
                                                                  ||                x                            |                                  
                                                                  ||                --                  for n = 0|                                  
                                                                  ||                2                            |                                  
  /                               //      x        for n = 0\     ||                                             |       //      x        for n = 0\
 |                                ||                        |     ||  //      /pi*n*x\               \           |       ||                        |
 |              /pi*n*x\          ||     /pi*n*x\           |     ||  ||-2*cos|------|               |           |       ||     /pi*n*x\           |
 | (2*x - 1)*cos|------| dx = C - |<2*sin|------|           | - 2*|<  ||      \  2   /      pi*n     |           | + 2*x*|<2*sin|------|           |
 |              \  2   /          ||     \  2   /           |     ||2*|<--------------  for ---- != 0|           |       ||     \  2   /           |
 |                                ||-------------  otherwise|     ||  ||     pi*n            2       |           |       ||-------------  otherwise|
/                                 \\     pi*n               /     ||  ||                             |           |       \\     pi*n               /
                                                                  ||  \\      0           otherwise  /           |                                  
                                                                  ||----------------------------------  otherwise|                                  
                                                                  ||               pi*n                          |                                  
                                                                  \\                                             /
∫(2x−1)cos⁡(xπn2) dx=C+2x({xfor n=02sin⁡(πnx2)πnotherwise)−{xfor n=02sin⁡(πnx2)πnotherwise−2({x22for n=02({−2cos⁡(πnx2)πnfor πn2≠00otherwise)πnotherwise)\int \left(2 x - 1\right) \cos{\left(\frac{x \pi n}{2} \right)}\, dx = C + 2 x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{2 \sin{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{otherwise} \end{cases}\right) - \begin{cases} x & \text{for}\: n = 0 \\\frac{2 \sin{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{otherwise} \end{cases} - 2 \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{2 \left(\begin{cases} - \frac{2 \cos{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{for}\: \frac{\pi n}{2} \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{\pi n} & \text{otherwise} \end{cases}\right)
Simplificar
Respuesta [src] 
/                /pi*n\                                  
|           8*cos|----|                                  
|    8           \ 4  /                                  
|- ------ + -----------  for And(n > -oo, n < oo, n != 0)
<    2  2        2  2                                    
|  pi *n       pi *n                                     
|                                                        
|         -1/4                      otherwise            
\
{8cos⁡(πn4)π2n2−8π2n2for n>−∞∧n<∞∧n≠0−14otherwise\begin{cases} \frac{8 \cos{\left(\frac{\pi n}{4} \right)}}{\pi^{2} n^{2}} - \frac{8}{\pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\- \frac{1}{4} & \text{otherwise} \end{cases} 
=
= 
/                /pi*n\                                  
|           8*cos|----|                                  
|    8           \ 4  /                                  
|- ------ + -----------  for And(n > -oo, n < oo, n != 0)
<    2  2        2  2                                    
|  pi *n       pi *n                                     
|                                                        
|         -1/4                      otherwise            
\
{8cos⁡(πn4)π2n2−8π2n2for n>−∞∧n<∞∧n≠0−14otherwise\begin{cases} \frac{8 \cos{\left(\frac{\pi n}{4} \right)}}{\pi^{2} n^{2}} - \frac{8}{\pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\- \frac{1}{4} & \text{otherwise} \end{cases} 
Piecewise((-8/(pi^2*n^2) + 8*cos(pi*n/4)/(pi^2*n^2), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (-1/4, True))

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

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