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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                   
  /                   
 |                    
 |       /2*pi*n*x\   
 |  x*cos|--------| dx
 |       \   t    /   
 |                    
/                     
0
∫01xcos⁡(x2πnt) dx\int\limits_{0}^{1} x \cos{\left(\frac{x 2 \pi n}{t} \right)}\, dx 
Integral(x*cos((((2*pi)*n)*x)/t), (x, 0, 1))
Respuesta (Indefinida) [src] 
                            //                   2                              \                                  
                            ||                  x                               |                                  
                            ||                  --                     for n = 0|                                  
                            ||                  2                               |                                  
  /                         ||                                                  |     //       x         for n = 0\
 |                          ||  //      /2*pi*n*x\                  \           |     ||                          |
 |      /2*pi*n*x\          ||  ||-t*cos|--------|                  |           |     ||     /2*pi*n*x\           |
 | x*cos|--------| dx = C - |<  ||      \   t    /       2*pi*n     |           | + x*|
∫xcos⁡(x2πnt) dx=C+x({xfor n=0tsin⁡(2πnxt)2πnotherwise)−{x22for n=0t({−tcos⁡(2πnxt)2πnfor 2πnt≠00otherwise)2πnotherwise\int x \cos{\left(\frac{x 2 \pi n}{t} \right)}\, dx = C + x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{t \sin{\left(\frac{2 \pi n x}{t} \right)}}{2 \pi n} & \text{otherwise} \end{cases}\right) - \begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{t \left(\begin{cases} - \frac{t \cos{\left(\frac{2 \pi n x}{t} \right)}}{2 \pi n} & \text{for}\: \frac{2 \pi n}{t} \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{2 \pi n} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/                  /2*pi*n\    2    /2*pi*n\                                  
|      2      t*sin|------|   t *cos|------|                                  
|     t            \  t   /         \  t   /                                  
|- -------- + ------------- + --------------  for And(n > -oo, n < oo, n != 0)
<      2  2       2*pi*n             2  2                                     
|  4*pi *n                       4*pi *n                                      
|                                                                             
|                    1/2                                 otherwise            
\
{tsin⁡(2πnt)2πn+t2cos⁡(2πnt)4π2n2−t24π2n2for n>−∞∧n<∞∧n≠012otherwise\begin{cases} \frac{t \sin{\left(\frac{2 \pi n}{t} \right)}}{2 \pi n} + \frac{t^{2} \cos{\left(\frac{2 \pi n}{t} \right)}}{4 \pi^{2} n^{2}} - \frac{t^{2}}{4 \pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{1}{2} & \text{otherwise} \end{cases} 
=
= 
/                  /2*pi*n\    2    /2*pi*n\                                  
|      2      t*sin|------|   t *cos|------|                                  
|     t            \  t   /         \  t   /                                  
|- -------- + ------------- + --------------  for And(n > -oo, n < oo, n != 0)
<      2  2       2*pi*n             2  2                                     
|  4*pi *n                       4*pi *n                                      
|                                                                             
|                    1/2                                 otherwise            
\
{tsin⁡(2πnt)2πn+t2cos⁡(2πnt)4π2n2−t24π2n2for n>−∞∧n<∞∧n≠012otherwise\begin{cases} \frac{t \sin{\left(\frac{2 \pi n}{t} \right)}}{2 \pi n} + \frac{t^{2} \cos{\left(\frac{2 \pi n}{t} \right)}}{4 \pi^{2} n^{2}} - \frac{t^{2}}{4 \pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{1}{2} & \text{otherwise} \end{cases} 
Piecewise((-t^2/(4*pi^2*n^2) + t*sin(2*pi*n/t)/(2*pi*n) + t^2*cos(2*pi*n/t)/(4*pi^2*n^2), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (1/2, True))

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

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