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Solving integrals/ (pi-x/2)*cos(n*x)

Integral of (pi-x/2)*cos(n*x) dx

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The solution

You have entered [src] 
 pi                     
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 3                      
  /                     
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 |  /     x\            
 |  |pi - -|*cos(n*x) dx
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pi
∫ππ3(−x2+π)cos⁡(nx) dx\int\limits_{\pi}^{\frac{\pi}{3}} \left(- \frac{x}{2} + \pi\right) \cos{\left(n x \right)}\, dx 
Integral((pi - x/2)*cos(n*x), (x, pi, pi/3))
The answer (Indefinite) [src] 
                              /           2                                                                             
                              |          x                                                                              
                              |          --             for n = 0                                                       
                              |          2                                                                              
                              |                                                                                         
                              |/-cos(n*x)                                                                               
                              <|----------  for n != 0                                                                  
                              |<    n                                                                                   
                              ||                                                                  //   x      for n = 0\
                              |\    0       otherwise                                             ||                   |
  /                           |-----------------------  otherwise                               x*|
∫(−x2+π)cos⁡(nx) dx=C−x({xfor n=0sin⁡(nx)notherwise)2+π({xfor n=0sin⁡(nx)notherwise)+{x22for n=0{−cos⁡(nx)nfor n≠00otherwisenotherwise2\int \left(- \frac{x}{2} + \pi\right) \cos{\left(n x \right)}\, dx = C - \frac{x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right)}{2} + \pi \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right) + \frac{\begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{\begin{cases} - \frac{\cos{\left(n x \right)}}{n} & \text{for}\: n \neq 0 \\0 & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}}{2}
Simplify
The answer [src] 
/               /pi*n\                          /pi*n\                                  
|            cos|----|                  5*pi*sin|----|                                  
|cos(pi*n)      \ 3  /   pi*sin(pi*n)           \ 3  /                                  
|--------- - --------- - ------------ + --------------  for And(n > -oo, n < oo, n != 0)
|      2           2         2*n             6*n                                        
<   2*n         2*n                                                                     
|                                                                                       
|                            2                                                          
|                       -4*pi                                                           
|                       ------                                     otherwise            
\                         9
{5πsin⁡(πn3)6n−πsin⁡(πn)2n−cos⁡(πn3)2n2+cos⁡(πn)2n2for n>−∞∧n<∞∧n≠0−4π29otherwise\begin{cases} \frac{5 \pi \sin{\left(\frac{\pi n}{3} \right)}}{6 n} - \frac{\pi \sin{\left(\pi n \right)}}{2 n} - \frac{\cos{\left(\frac{\pi n}{3} \right)}}{2 n^{2}} + \frac{\cos{\left(\pi n \right)}}{2 n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\- \frac{4 \pi^{2}}{9} & \text{otherwise} \end{cases} 
=
= 
/               /pi*n\                          /pi*n\                                  
|            cos|----|                  5*pi*sin|----|                                  
|cos(pi*n)      \ 3  /   pi*sin(pi*n)           \ 3  /                                  
|--------- - --------- - ------------ + --------------  for And(n > -oo, n < oo, n != 0)
|      2           2         2*n             6*n                                        
<   2*n         2*n                                                                     
|                                                                                       
|                            2                                                          
|                       -4*pi                                                           
|                       ------                                     otherwise            
\                         9
{5πsin⁡(πn3)6n−πsin⁡(πn)2n−cos⁡(πn3)2n2+cos⁡(πn)2n2for n>−∞∧n<∞∧n≠0−4π29otherwise\begin{cases} \frac{5 \pi \sin{\left(\frac{\pi n}{3} \right)}}{6 n} - \frac{\pi \sin{\left(\pi n \right)}}{2 n} - \frac{\cos{\left(\frac{\pi n}{3} \right)}}{2 n^{2}} + \frac{\cos{\left(\pi n \right)}}{2 n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\- \frac{4 \pi^{2}}{9} & \text{otherwise} \end{cases} 
Piecewise((cos(pi*n)/(2*n^2) - cos(pi*n/3)/(2*n^2) - pi*sin(pi*n)/(2*n) + 5*pi*sin(pi*n/3)/(6*n), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (-4*pi^2/9, True))

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

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