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Resolución de integrales/ sin(npix)(x-x²)/(npi)

Integral de sin(npix)(x-x²)/(npi) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                        
  /                        
 |                         
 |              /     2\   
 |  sin(n*pi*x)*\x - x /   
 |  -------------------- dx
 |          n*pi           
 |                         
/                          
0
∫01(−x2+x)sin⁡(xπn)πn dx\int\limits_{0}^{1} \frac{\left(- x^{2} + x\right) \sin{\left(x \pi n \right)}}{\pi n}\, dx 
Integral((sin((n*pi)*x)*(x - x^2))/((n*pi)), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                      /                                                    //                     0                        for n = 0\                                                                 \
                                      |                                                    ||                                                       |                                                                 |
  /                                   |  //               0                 for n = 0\     || //cos(pi*n*x)   x*sin(pi*n*x)            \            |                                                                 |
 |                                    |  ||                                          |     || ||----------- + -------------  for n != 0|            |                                                                 |
 |             /     2\               |  || //sin(pi*n*x)               \            |     || ||     2  2          pi*n                |            |     //      0        for n = 0\      //      0        for n = 0\|
 | sin(n*pi*x)*\x - x /           1   |  || ||-----------  for pi*n != 0|            |     || ||   pi *n                               |            |     ||                        |    2 ||                        ||
 | -------------------- dx = C + ----*|- |<-|<    pi*n                  |            | + 2*|<-|<                                       |            | + x*|<-cos(pi*n*x)            | - x *|<-cos(pi*n*x)            ||
 |         n*pi                  pi*n |  || ||                          |            |     || ||             2                         |            |     ||-------------  otherwise|      ||-------------  otherwise||
 |                                    |  || \\     x         otherwise  /            |     || ||            x                          |            |     \\     pi*n               /      \\     pi*n               /|
/                                     |  ||-------------------------------  otherwise|     || ||            --               otherwise |            |                                                                 |
                                      |  \\              pi*n                        /     || \\            2                          /            |                                                                 |
                                      |                                                    ||--------------------------------------------  otherwise|                                                                 |
                                      \                                                    \\                    pi*n                               /                                                                 /
∫(−x2+x)sin⁡(xπn)πn dx=C+1πn(−x2({0for n=0−cos⁡(πnx)πnotherwise)+x({0for n=0−cos⁡(πnx)πnotherwise)−{0for n=0−{sin⁡(πnx)πnfor πn≠0xotherwiseπnotherwise+2({0for n=0−{xsin⁡(πnx)πn+cos⁡(πnx)π2n2for n≠0x22otherwiseπnotherwise))\int \frac{\left(- x^{2} + x\right) \sin{\left(x \pi n \right)}}{\pi n}\, dx = C + \frac{1}{\pi n} \left(- x^{2} \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\cos{\left(\pi n x \right)}}{\pi n} & \text{otherwise} \end{cases}\right) + x \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\cos{\left(\pi n x \right)}}{\pi n} & \text{otherwise} \end{cases}\right) - \begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\begin{cases} \frac{\sin{\left(\pi n x \right)}}{\pi n} & \text{for}\: \pi n \neq 0 \\x & \text{otherwise} \end{cases}}{\pi n} & \text{otherwise} \end{cases} + 2 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\begin{cases} \frac{x \sin{\left(\pi n x \right)}}{\pi n} + \frac{\cos{\left(\pi n x \right)}}{\pi^{2} n^{2}} & \text{for}\: n \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases}}{\pi n} & \text{otherwise} \end{cases}\right)\right)
Simplificar
Respuesta [src] 
/           sin(pi*n)   2*cos(pi*n)                                  
|         - --------- - -----------                                  
|               2  2         3  3                                    
|  2          pi *n        pi *n                                     
<------ + -------------------------  for And(n > -oo, n < oo, n != 0)
|  4  4              pi*n                                            
|pi *n                                                               
|                                                                    
\                0                              otherwise
{−sin⁡(πn)π2n2−2cos⁡(πn)π3n3πn+2π4n4for n>−∞∧n<∞∧n≠00otherwise\begin{cases} \frac{- \frac{\sin{\left(\pi n \right)}}{\pi^{2} n^{2}} - \frac{2 \cos{\left(\pi n \right)}}{\pi^{3} n^{3}}}{\pi n} + \frac{2}{\pi^{4} n^{4}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases} 
=
= 
/           sin(pi*n)   2*cos(pi*n)                                  
|         - --------- - -----------                                  
|               2  2         3  3                                    
|  2          pi *n        pi *n                                     
<------ + -------------------------  for And(n > -oo, n < oo, n != 0)
|  4  4              pi*n                                            
|pi *n                                                               
|                                                                    
\                0                              otherwise
{−sin⁡(πn)π2n2−2cos⁡(πn)π3n3πn+2π4n4for n>−∞∧n<∞∧n≠00otherwise\begin{cases} \frac{- \frac{\sin{\left(\pi n \right)}}{\pi^{2} n^{2}} - \frac{2 \cos{\left(\pi n \right)}}{\pi^{3} n^{3}}}{\pi n} + \frac{2}{\pi^{4} n^{4}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases} 
Piecewise((2/(pi^4*n^4) + (-sin(pi*n)/(pi^2*n^2) - 2*cos(pi*n)/(pi^3*n^3))/(pi*n), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (0, True))

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

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