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Resolución de integrales/ cos(πx/2l)x

Integral de cos(πx/2l)x dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  0                 
  /                 
 |                  
 |     /pi*x  \     
 |  cos|----*l|*x dx
 |     \ 2    /     
 |                  
/                   
l                   
-                   
2
$$\int\limits_{\frac{l}{2}}^{0} x \cos{\left(l \frac{\pi x}{2} \right)}\, dx$$ 
Integral(cos(((pi*x)/2)*l)*x, (x, l/2, 0))
Respuesta (Indefinida) [src] 
                          //                 2                           \                                
                          ||                x                            |                                
                          ||                --                  for l = 0|                                
                          ||                2                            |                                
  /                       ||                                             |     //      x        for l = 0\
 |                        ||  //      /pi*l*x\               \           |     ||                        |
 |    /pi*x  \            ||  ||-2*cos|------|               |           |     ||     /pi*l*x\           |
 | cos|----*l|*x dx = C - |<  ||      \  2   /      pi*l     |           | + x*|<2*sin|------|           |
 |    \ 2    /            ||2*|<--------------  for ---- != 0|           |     ||     \  2   /           |
 |                        ||  ||     pi*l            2       |           |     ||-------------  otherwise|
/                         ||  ||                             |           |     \\     pi*l               /
                          ||  \\      0           otherwise  /           |                                
                          ||----------------------------------  otherwise|                                
                          ||               pi*l                          |                                
                          \\                                             /
$$\int x \cos{\left(l \frac{\pi x}{2} \right)}\, dx = C + x \left(\begin{cases} x & \text{for}\: l = 0 \\\frac{2 \sin{\left(\frac{\pi l x}{2} \right)}}{\pi l} & \text{otherwise} \end{cases}\right) - \begin{cases} \frac{x^{2}}{2} & \text{for}\: l = 0 \\\frac{2 \left(\begin{cases} - \frac{2 \cos{\left(\frac{\pi l x}{2} \right)}}{\pi l} & \text{for}\: \frac{\pi l}{2} \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{\pi l} & \text{otherwise} \end{cases}$$
Simplificar
Respuesta [src] 
/     /    2\                 /    2\                                  
|     |pi*l |                 |pi*l |                                  
|  sin|-----|            4*cos|-----|                                  
|     \  4  /     4           \  4  /                                  
|- ---------- + ------ - ------------  for And(l > -oo, l < oo, l != 0)
|      pi         2  2        2  2                                     
<               pi *l       pi *l                                      
|                                                                      
|                  2                                                   
|                -l                                                    
|                ----                             otherwise            
|                 8                                                    
\
$$\begin{cases} - \frac{\sin{\left(\frac{\pi l^{2}}{4} \right)}}{\pi} - \frac{4 \cos{\left(\frac{\pi l^{2}}{4} \right)}}{\pi^{2} l^{2}} + \frac{4}{\pi^{2} l^{2}} & \text{for}\: l > -\infty \wedge l < \infty \wedge l \neq 0 \\- \frac{l^{2}}{8} & \text{otherwise} \end{cases}$$ 
=
= 
/     /    2\                 /    2\                                  
|     |pi*l |                 |pi*l |                                  
|  sin|-----|            4*cos|-----|                                  
|     \  4  /     4           \  4  /                                  
|- ---------- + ------ - ------------  for And(l > -oo, l < oo, l != 0)
|      pi         2  2        2  2                                     
<               pi *l       pi *l                                      
|                                                                      
|                  2                                                   
|                -l                                                    
|                ----                             otherwise            
|                 8                                                    
\
$$\begin{cases} - \frac{\sin{\left(\frac{\pi l^{2}}{4} \right)}}{\pi} - \frac{4 \cos{\left(\frac{\pi l^{2}}{4} \right)}}{\pi^{2} l^{2}} + \frac{4}{\pi^{2} l^{2}} & \text{for}\: l > -\infty \wedge l < \infty \wedge l \neq 0 \\- \frac{l^{2}}{8} & \text{otherwise} \end{cases}$$ 
Piecewise((-sin(pi*l^2/4)/pi + 4/(pi^2*l^2) - 4*cos(pi*l^2/4)/(pi^2*l^2), (l > -oo)∧(l < oo)∧(Ne(l, 0))), (-l^2/8, True))

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

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