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

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The solution

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 pi                     
  /                     
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 |  x*sin(x)*cos(n*x) dx
 |                      
/                       
-pi                     
$$\int\limits_{- \pi}^{\pi} x \sin{\left(x \right)} \cos{\left(n x \right)}\, dx$$
Integral((x*sin(x))*cos(n*x), (x, -pi, pi))
The answer (Indefinite) [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                           |
                              \\                                                                                                                                                                         /
$$\int x \sin{\left(x \right)} \cos{\left(n x \right)}\, dx = C + \begin{cases} \frac{x \sin^{2}{\left(x \right)}}{4} - \frac{x \cos^{2}{\left(x \right)}}{4} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{4} & \text{for}\: n = -1 \vee n = 1 \\\frac{n^{3} x \sin{\left(x \right)} \sin{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} + \frac{n^{2} x \cos{\left(x \right)} \cos{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} + \frac{n^{2} \sin{\left(x \right)} \cos{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} - \frac{n x \sin{\left(x \right)} \sin{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} - \frac{2 n \sin{\left(n x \right)} \cos{\left(x \right)}}{n^{4} - 2 n^{2} + 1} - \frac{x \cos{\left(x \right)} \cos{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} + \frac{\sin{\left(x \right)} \cos{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} & \text{otherwise} \end{cases}$$
The answer [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                          
$$\begin{cases} - \frac{\pi}{2} & \text{for}\: n = -1 \vee n = 1 \\- \frac{2 \pi n^{2} \cos{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{4 n \sin{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{2 \pi \cos{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} & \text{otherwise} \end{cases}$$
=
=
/                       -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                          
$$\begin{cases} - \frac{\pi}{2} & \text{for}\: n = -1 \vee n = 1 \\- \frac{2 \pi n^{2} \cos{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{4 n \sin{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} + \frac{2 \pi \cos{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} & \text{otherwise} \end{cases}$$
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))

    Use the examples entering the upper and lower limits of integration.