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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  l                        
  /                        
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 |  2                      
 |  --*x*cos(x)*sin(n*x) dx
 |  pi                     
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/                          
0
$$\int\limits_{0}^{l} \frac{2}{\pi} x \cos{\left(x \right)} \sin{\left(n x \right)}\, dx$$ 
Integral((((2/pi)*x)*cos(x))*sin(n*x), (x, 0, l))
Respuesta (Indefinida) [src] 
                                 //                                                        /       2           2                   \                                                                 \
                                 ||                                                        |  x*cos (x)   x*sin (x)   cos(x)*sin(x)|                                                                 |
                                 ||                                                     -2*|- --------- + --------- + -------------|                                                                 |
                                 ||                                                        \      4           4             4      /                                                                 |
                                 ||                                                     --------------------------------------------                                                       for n = -1|
                                 ||                                                                          pi                                                                                      |
                                 ||                                                                                                                                                                  |
  /                              ||                                                       /       2           2                   \                                                                  |
 |                               ||                                                       |  x*cos (x)   x*sin (x)   cos(x)*sin(x)|                                                                  |
 | 2                             ||                                                     2*|- --------- + --------- + -------------|                                                                  |
 | --*x*cos(x)*sin(n*x) dx = C + |<                                                       \      4           4             4      /                                                                  |
 | pi                            ||                                                     -------------------------------------------                                                        for n = 1 |
 |                               ||                                                                          pi                                                                                      |
/                                ||                                                                                                                                                                  |
                                 ||  /                                       2                                                                  2                      3                \            |
                                 ||  |cos(x)*sin(n*x)   x*sin(x)*sin(n*x)   n *cos(x)*sin(n*x)   2*n*cos(n*x)*sin(x)   n*x*cos(x)*cos(n*x)   x*n *sin(x)*sin(n*x)   x*n *cos(x)*cos(n*x)|            |
                                 ||2*|--------------- + ----------------- + ------------------ - ------------------- + ------------------- - -------------------- - --------------------|            |
                                 ||  |      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     /            |
                                 ||------------------------------------------------------------------------------------------------------------------------------------------------------  otherwise |
                                 \\                                                                          pi                                                                                      /
$$\int \frac{2}{\pi} x \cos{\left(x \right)} \sin{\left(n x \right)}\, dx = C + \begin{cases} - \frac{2 \left(\frac{x \sin^{2}{\left(x \right)}}{4} - \frac{x \cos^{2}{\left(x \right)}}{4} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{4}\right)}{\pi} & \text{for}\: n = -1 \\\frac{2 \left(\frac{x \sin^{2}{\left(x \right)}}{4} - \frac{x \cos^{2}{\left(x \right)}}{4} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{4}\right)}{\pi} & \text{for}\: n = 1 \\\frac{2 \left(- \frac{n^{3} x \cos{\left(x \right)} \cos{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} - \frac{n^{2} x \sin{\left(x \right)} \sin{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} + \frac{n^{2} \sin{\left(n x \right)} \cos{\left(x \right)}}{n^{4} - 2 n^{2} + 1} + \frac{n x \cos{\left(x \right)} \cos{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} - \frac{2 n \sin{\left(x \right)} \cos{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} + \frac{x \sin{\left(x \right)} \sin{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} + \frac{\sin{\left(n x \right)} \cos{\left(x \right)}}{n^{4} - 2 n^{2} + 1}\right)}{\pi} & \text{otherwise} \end{cases}$$
Simplificar
Respuesta [src] 
/                                                       /       2                           2   \                                                                  
|                                                       |  l*sin (l)   cos(l)*sin(l)   l*cos (l)|                                                                  
|                                                     2*|- --------- - ------------- + ---------|                                                                  
|                                                       \      4             4             4    /                                                                  
|                                                     -------------------------------------------                                                        for n = -1
|                                                                          pi                                                                                      
|                                                                                                                                                                  
|                                                       /       2           2                   \                                                                  
|                                                       |  l*cos (l)   l*sin (l)   cos(l)*sin(l)|                                                                  
|                                                     2*|- --------- + --------- + -------------|                                                                  
<                                                       \      4           4             4      /                                                                  
|                                                     -------------------------------------------                                                        for n = 1 
|                                                                          pi                                                                                      
|                                                                                                                                                                  
|  /                                       2                                                                  2                      3                \            
|  |cos(l)*sin(l*n)   l*sin(l)*sin(l*n)   n *cos(l)*sin(l*n)   2*n*cos(l*n)*sin(l)   l*n*cos(l)*cos(l*n)   l*n *sin(l)*sin(l*n)   l*n *cos(l)*cos(l*n)|            
|2*|--------------- + ----------------- + ------------------ - ------------------- + ------------------- - -------------------- - --------------------|            
|  |      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     /            
|------------------------------------------------------------------------------------------------------------------------------------------------------  otherwise 
\                                                                          pi
$$\begin{cases} \frac{2 \left(- \frac{l \sin^{2}{\left(l \right)}}{4} + \frac{l \cos^{2}{\left(l \right)}}{4} - \frac{\sin{\left(l \right)} \cos{\left(l \right)}}{4}\right)}{\pi} & \text{for}\: n = -1 \\\frac{2 \left(\frac{l \sin^{2}{\left(l \right)}}{4} - \frac{l \cos^{2}{\left(l \right)}}{4} + \frac{\sin{\left(l \right)} \cos{\left(l \right)}}{4}\right)}{\pi} & \text{for}\: n = 1 \\\frac{2 \left(- \frac{l n^{3} \cos{\left(l \right)} \cos{\left(l n \right)}}{n^{4} - 2 n^{2} + 1} - \frac{l n^{2} \sin{\left(l \right)} \sin{\left(l n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{l n \cos{\left(l \right)} \cos{\left(l n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{l \sin{\left(l \right)} \sin{\left(l n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{n^{2} \sin{\left(l n \right)} \cos{\left(l \right)}}{n^{4} - 2 n^{2} + 1} - \frac{2 n \sin{\left(l \right)} \cos{\left(l n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{\sin{\left(l n \right)} \cos{\left(l \right)}}{n^{4} - 2 n^{2} + 1}\right)}{\pi} & \text{otherwise} \end{cases}$$ 
=
= 
/                                                       /       2                           2   \                                                                  
|                                                       |  l*sin (l)   cos(l)*sin(l)   l*cos (l)|                                                                  
|                                                     2*|- --------- - ------------- + ---------|                                                                  
|                                                       \      4             4             4    /                                                                  
|                                                     -------------------------------------------                                                        for n = -1
|                                                                          pi                                                                                      
|                                                                                                                                                                  
|                                                       /       2           2                   \                                                                  
|                                                       |  l*cos (l)   l*sin (l)   cos(l)*sin(l)|                                                                  
|                                                     2*|- --------- + --------- + -------------|                                                                  
<                                                       \      4           4             4      /                                                                  
|                                                     -------------------------------------------                                                        for n = 1 
|                                                                          pi                                                                                      
|                                                                                                                                                                  
|  /                                       2                                                                  2                      3                \            
|  |cos(l)*sin(l*n)   l*sin(l)*sin(l*n)   n *cos(l)*sin(l*n)   2*n*cos(l*n)*sin(l)   l*n*cos(l)*cos(l*n)   l*n *sin(l)*sin(l*n)   l*n *cos(l)*cos(l*n)|            
|2*|--------------- + ----------------- + ------------------ - ------------------- + ------------------- - -------------------- - --------------------|            
|  |      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     /            
|------------------------------------------------------------------------------------------------------------------------------------------------------  otherwise 
\                                                                          pi
$$\begin{cases} \frac{2 \left(- \frac{l \sin^{2}{\left(l \right)}}{4} + \frac{l \cos^{2}{\left(l \right)}}{4} - \frac{\sin{\left(l \right)} \cos{\left(l \right)}}{4}\right)}{\pi} & \text{for}\: n = -1 \\\frac{2 \left(\frac{l \sin^{2}{\left(l \right)}}{4} - \frac{l \cos^{2}{\left(l \right)}}{4} + \frac{\sin{\left(l \right)} \cos{\left(l \right)}}{4}\right)}{\pi} & \text{for}\: n = 1 \\\frac{2 \left(- \frac{l n^{3} \cos{\left(l \right)} \cos{\left(l n \right)}}{n^{4} - 2 n^{2} + 1} - \frac{l n^{2} \sin{\left(l \right)} \sin{\left(l n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{l n \cos{\left(l \right)} \cos{\left(l n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{l \sin{\left(l \right)} \sin{\left(l n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{n^{2} \sin{\left(l n \right)} \cos{\left(l \right)}}{n^{4} - 2 n^{2} + 1} - \frac{2 n \sin{\left(l \right)} \cos{\left(l n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{\sin{\left(l n \right)} \cos{\left(l \right)}}{n^{4} - 2 n^{2} + 1}\right)}{\pi} & \text{otherwise} \end{cases}$$ 
Piecewise((2*(-l*sin(l)^2/4 - cos(l)*sin(l)/4 + l*cos(l)^2/4)/pi, n = -1), (2*(-l*cos(l)^2/4 + l*sin(l)^2/4 + cos(l)*sin(l)/4)/pi, n = 1), (2*(cos(l)*sin(l*n)/(1 + n^4 - 2*n^2) + l*sin(l)*sin(l*n)/(1 + n^4 - 2*n^2) + n^2*cos(l)*sin(l*n)/(1 + n^4 - 2*n^2) - 2*n*cos(l*n)*sin(l)/(1 + n^4 - 2*n^2) + l*n*cos(l)*cos(l*n)/(1 + n^4 - 2*n^2) - l*n^2*sin(l)*sin(l*n)/(1 + n^4 - 2*n^2) - l*n^3*cos(l)*cos(l*n)/(1 + n^4 - 2*n^2))/pi, True))

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

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