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Resolución de integrales/ (x^2)/ i*n*x/ (2*(x^2)+x)*cos(pi*n*x)

Integral de (2*(x^2)+x)*cos(pi*n*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                          
  /                          
 |                           
 |  /   2    \               
 |  \2*x  + x/*cos(pi*n*x) dx
 |                           
/                            
-1
∫−11(2x2+x)cos⁡(xπn) dx\int\limits_{-1}^{1} \left(2 x^{2} + x\right) \cos{\left(x \pi n \right)}\, dx 
Integral((2*x^2 + x)*cos((pi*n)*x), (x, -1, 1))
Respuesta (Indefinida) [src] 
                                                                                   //                    3                              \                                                               
                                   //              2                         \     ||                   x                               |                                                               
                                   ||             x                          |     ||                   --                     for n = 0|                                                               
                                   ||             --                for n = 0|     ||                   3                               |                                                               
  /                                ||             2                          |     ||                                                   |                                                               
 |                                 ||                                        |     ||/sin(pi*n*x)   x*cos(pi*n*x)                       |     //     x       for n = 0\        //     x       for n = 0\
 | /   2    \                      ||/-cos(pi*n*x)                           |     |||----------- - -------------  for n != 0           |     ||                      |      2 ||                      |
 | \2*x  + x/*cos(pi*n*x) dx = C - |<|-------------  for pi*n != 0           | - 4*|<|     2  2          pi*n                           | + x*|
∫(2x2+x)cos⁡(xπn) dx=C+2x2({xfor n=0sin⁡(πnx)πnotherwise)+x({xfor n=0sin⁡(πnx)πnotherwise)−{x22for n=0{−cos⁡(πnx)πnfor πn≠00otherwiseπnotherwise−4({x33for n=0{−xcos⁡(πnx)πn+sin⁡(πnx)π2n2for n≠00otherwiseπnotherwise)\int \left(2 x^{2} + x\right) \cos{\left(x \pi n \right)}\, dx = C + 2 x^{2} \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(\pi n x \right)}}{\pi n} & \text{otherwise} \end{cases}\right) + x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(\pi n x \right)}}{\pi n} & \text{otherwise} \end{cases}\right) - \begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{\begin{cases} - \frac{\cos{\left(\pi n x \right)}}{\pi n} & \text{for}\: \pi n \neq 0 \\0 & \text{otherwise} \end{cases}}{\pi n} & \text{otherwise} \end{cases} - 4 \left(\begin{cases} \frac{x^{3}}{3} & \text{for}\: n = 0 \\\frac{\begin{cases} - \frac{x \cos{\left(\pi n x \right)}}{\pi n} + \frac{\sin{\left(\pi n x \right)}}{\pi^{2} n^{2}} & \text{for}\: n \neq 0 \\0 & \text{otherwise} \end{cases}}{\pi n} & \text{otherwise} \end{cases}\right)
Simplificar
Respuesta [src] 
/  8*sin(pi*n)   4*sin(pi*n)   8*cos(pi*n)                                  
|- ----------- + ----------- + -----------  for And(n > -oo, n < oo, n != 0)
|       3  3         pi*n           2  2                                    
<     pi *n                       pi *n                                     
|                                                                           
|                   4/3                                otherwise            
\
{4sin⁡(πn)πn+8cos⁡(πn)π2n2−8sin⁡(πn)π3n3for n>−∞∧n<∞∧n≠043otherwise\begin{cases} \frac{4 \sin{\left(\pi n \right)}}{\pi n} + \frac{8 \cos{\left(\pi n \right)}}{\pi^{2} n^{2}} - \frac{8 \sin{\left(\pi n \right)}}{\pi^{3} n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{4}{3} & \text{otherwise} \end{cases} 
=
= 
/  8*sin(pi*n)   4*sin(pi*n)   8*cos(pi*n)                                  
|- ----------- + ----------- + -----------  for And(n > -oo, n < oo, n != 0)
|       3  3         pi*n           2  2                                    
<     pi *n                       pi *n                                     
|                                                                           
|                   4/3                                otherwise            
\
{4sin⁡(πn)πn+8cos⁡(πn)π2n2−8sin⁡(πn)π3n3for n>−∞∧n<∞∧n≠043otherwise\begin{cases} \frac{4 \sin{\left(\pi n \right)}}{\pi n} + \frac{8 \cos{\left(\pi n \right)}}{\pi^{2} n^{2}} - \frac{8 \sin{\left(\pi n \right)}}{\pi^{3} n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{4}{3} & \text{otherwise} \end{cases} 
Piecewise((-8*sin(pi*n)/(pi^3*n^3) + 4*sin(pi*n)/(pi*n) + 8*cos(pi*n)/(pi^2*n^2), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (4/3, True))

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

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