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Resolución de integrales/ xcos(2nx/a)

Integral de xcos(2nx/a) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

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

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