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  • Identical expressions

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  • Similar expressions

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

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

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src] 
 pi                       
  /                       
 |                        
 |  /  2    2\            
 |  \pi  - x /*cos(n*x) dx
 |                        
/                         
-pi
$$\int\limits_{- \pi}^{\pi} \left(- x^{2} + \pi^{2}\right) \cos{\left(n x \right)}\, dx$$ 
Integral((pi^2 - x^2)*cos(n*x), (x, -pi, pi))
The answer (Indefinite) [src] 
                                  //                 3                           \                                                         
                                  ||                x                            |                                                         
                                  ||                --                  for n = 0|                                                         
                                  ||                3                            |                                                         
  /                               ||                                             |                                                         
 |                                ||/sin(n*x)   x*cos(n*x)                       |       //   x      for n = 0\      //   x      for n = 0\
 | /  2    2\                     |||-------- - ----------  for n != 0           |     2 ||                   |    2 ||                   |
 | \pi  - x /*cos(n*x) dx = C + 2*|<|    2          n                            | + pi *|
$${{\pi^2\,\sin \left(n\,x\right)-{{\left(n^2\,x^2-2\right)\,\sin \left(n\,x\right)+2\,n\,x\,\cos \left(n\,x\right)}\over{n^2}}}\over{ n}}$$
Simplify
The answer [src] 
/4*sin(pi*n)   4*pi*cos(pi*n)                                  
|----------- - --------------  for And(n > -oo, n < oo, n != 0)
|      3              2                                        
|     n              n                                         
<                                                              
|               3                                              
|           4*pi                                               
|           -----                         otherwise            
\             3
$${{2\,\left(2\,\sin \left(n\,\pi\right)-2\,n\,\pi\,\cos \left(n\,\pi \right)\right)}\over{n^3}}$$ 
=
= 
/4*sin(pi*n)   4*pi*cos(pi*n)                                  
|----------- - --------------  for And(n > -oo, n < oo, n != 0)
|      3              2                                        
|     n              n                                         
<                                                              
|               3                                              
|           4*pi                                               
|           -----                         otherwise            
\             3
$$\begin{cases} - \frac{4 \pi \cos{\left(\pi n \right)}}{n^{2}} + \frac{4 \sin{\left(\pi n \right)}}{n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{4 \pi^{3}}{3} & \text{otherwise} \end{cases}$$
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