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Resolución de integrales/ cos(n*x)/ sin(x)/ x*sin(x)cos(n*x)

Integral de x*sin(x)cos(n*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi                     
  /                     
 |                      
 |  x*sin(x)*cos(n*x) dx
 |                      
/                       
-pi
∫−ππxsin⁡(x)cos⁡(nx) dx\int\limits_{- \pi}^{\pi} x \sin{\left(x \right)} \cos{\left(n x \right)}\, dx 
Integral((x*sin(x))*cos(n*x), (x, -pi, pi))
Respuesta (Indefinida) [src] 
                              //                                                            2           2                                                                                                \
                              ||                                                       x*cos (x)   x*sin (x)   cos(x)*sin(x)                                                                             |
                              ||                                                     - --------- + --------- + -------------                                                        for Or(n = -1, n = 1)|
  /                           ||                                                           4           4             4                                                                                   |
 |                            ||                                                                                                                                                                         |
 | x*sin(x)*cos(n*x) dx = C + |<                   2                                                                2                      3                                                             |
 |                            ||cos(n*x)*sin(x)   n *cos(n*x)*sin(x)   x*cos(x)*cos(n*x)   2*n*cos(x)*sin(n*x)   x*n *cos(x)*cos(n*x)   x*n *sin(x)*sin(n*x)   n*x*sin(x)*sin(n*x)                       |
/                             ||--------------- + ------------------ - ----------------- - ------------------- + -------------------- + -------------------- - -------------------        otherwise      |
                              ||      4      2           4      2             4      2             4      2              4      2               4      2               4      2                          |
                              || 1 + n  - 2*n       1 + n  - 2*n         1 + n  - 2*n         1 + n  - 2*n          1 + n  - 2*n           1 + n  - 2*n           1 + n  - 2*n                           |
                              \\                                                                                                                                                                         /
∫xsin⁡(x)cos⁡(nx) dx=C+{xsin⁡2(x)4−xcos⁡2(x)4+sin⁡(x)cos⁡(x)4for n=−1∨n=1n3xsin⁡(x)sin⁡(nx)n4−2n2+1+n2xcos⁡(x)cos⁡(nx)n4−2n2+1+n2sin⁡(x)cos⁡(nx)n4−2n2+1−nxsin⁡(x)sin⁡(nx)n4−2n2+1−2nsin⁡(nx)cos⁡(x)n4−2n2+1−xcos⁡(x)cos⁡(nx)n4−2n2+1+sin⁡(x)cos⁡(nx)n4−2n2+1otherwise\int x \sin{\left(x \right)} \cos{\left(n x \right)}\, dx = C + \begin{cases} \frac{x \sin^{2}{\left(x \right)}}{4} - \frac{x \cos^{2}{\left(x \right)}}{4} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{4} & \text{for}\: n = -1 \vee n = 1 \\\frac{n^{3} x \sin{\left(x \right)} \sin{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} + \frac{n^{2} x \cos{\left(x \right)} \cos{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} + \frac{n^{2} \sin{\left(x \right)} \cos{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} - \frac{n x \sin{\left(x \right)} \sin{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} - \frac{2 n \sin{\left(n x \right)} \cos{\left(x \right)}}{n^{4} - 2 n^{2} + 1} - \frac{x \cos{\left(x \right)} \cos{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} + \frac{\sin{\left(x \right)} \cos{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/                       -pi                                               
|                       ----                         for Or(n = -1, n = 1)
|                        2                                                
|                                                                         
<                                       2                                 
|2*pi*cos(pi*n)   4*n*sin(pi*n)   2*pi*n *cos(pi*n)                       
|-------------- + ------------- - -----------------        otherwise      
|     4      2         4      2          4      2                         
\1 + n  - 2*n     1 + n  - 2*n      1 + n  - 2*n
{−π2for n=−1∨n=1−2πn2cos⁡(πn)n4−2n2+1+4nsin⁡(πn)n4−2n2+1+2πcos⁡(πn)n4−2n2+1otherwise\begin{cases} - \frac{\pi}{2} & \text{for}\: n = -1 \vee n = 1 \\- \frac{2 \pi n^{2} \cos{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{4 n \sin{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{2 \pi \cos{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} & \text{otherwise} \end{cases} 
=
= 
/                       -pi                                               
|                       ----                         for Or(n = -1, n = 1)
|                        2                                                
|                                                                         
<                                       2                                 
|2*pi*cos(pi*n)   4*n*sin(pi*n)   2*pi*n *cos(pi*n)                       
|-------------- + ------------- - -----------------        otherwise      
|     4      2         4      2          4      2                         
\1 + n  - 2*n     1 + n  - 2*n      1 + n  - 2*n
{−π2for n=−1∨n=1−2πn2cos⁡(πn)n4−2n2+1+4nsin⁡(πn)n4−2n2+1+2πcos⁡(πn)n4−2n2+1otherwise\begin{cases} - \frac{\pi}{2} & \text{for}\: n = -1 \vee n = 1 \\- \frac{2 \pi n^{2} \cos{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{4 n \sin{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{2 \pi \cos{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} & \text{otherwise} \end{cases} 
Piecewise((-pi/2, (n = -1)∨(n = 1)), (2*pi*cos(pi*n)/(1 + n^4 - 2*n^2) + 4*n*sin(pi*n)/(1 + n^4 - 2*n^2) - 2*pi*n^2*cos(pi*n)/(1 + n^4 - 2*n^2), True))

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

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