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Integral of (-1-x)cos((npix)/2) dx

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  2                        
  /                        
 |                         
 |              /n*pi*x\   
 |  (-1 - x)*cos|------| dx
 |              \  2   /   
 |                         
/                          
-2                         
$$\int\limits_{-2}^{2} \left(- x - 1\right) \cos{\left(\frac{\pi n x}{2} \right)}\, dx$$
Integral((-1 - x)*cos(n*pi*x/2), (x, -2, 2))
The answer (Indefinite) [src]
                                                                                               //                 2                           \
                                                                                               ||                x                            |
                                                                                               ||                --                  for n = 0|
                                                                                               ||                2                            |
  /                              //      x        for n = 0\     //      x        for n = 0\   ||                                             |
 |                               ||                        |     ||                        |   ||  //      /pi*n*x\               \           |
 |             /n*pi*x\          ||     /pi*n*x\           |     ||     /pi*n*x\           |   ||  ||-2*cos|------|               |           |
 | (-1 - x)*cos|------| dx = C - |<2*sin|------|           | - x*|<2*sin|------|           | + |<  ||      \  2   /               |           |
 |             \  2   /          ||     \  2   /           |     ||     \  2   /           |   ||2*|<--------------  for pi*n != 0|           |
 |                               ||-------------  otherwise|     ||-------------  otherwise|   ||  ||     pi*n                    |           |
/                                \\     pi*n               /     \\     pi*n               /   ||  ||                             |           |
                                                                                               ||  \\      0           otherwise  /           |
                                                                                               ||----------------------------------  otherwise|
                                                                                               ||               pi*n                          |
                                                                                               \\                                             /
$$\int \left(- x - 1\right) \cos{\left(\frac{\pi n x}{2} \right)}\, dx = C - x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{2 \sin{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{otherwise} \end{cases}\right) - \begin{cases} x & \text{for}\: n = 0 \\\frac{2 \sin{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{otherwise} \end{cases} + \begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{2 \left(\begin{cases} - \frac{2 \cos{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{for}\: \pi n \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{\pi n} & \text{otherwise} \end{cases}$$
The answer [src]
/-4*sin(pi*n)                                  
|------------  for And(n > -oo, n < oo, n != 0)
<    pi*n                                      
|                                              
\     -4                  otherwise            
$$\begin{cases} - \frac{4 \sin{\left(\pi n \right)}}{\pi n} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\-4 & \text{otherwise} \end{cases}$$
=
=
/-4*sin(pi*n)                                  
|------------  for And(n > -oo, n < oo, n != 0)
<    pi*n                                      
|                                              
\     -4                  otherwise            
$$\begin{cases} - \frac{4 \sin{\left(\pi n \right)}}{\pi n} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\-4 & \text{otherwise} \end{cases}$$

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