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Resolución de integrales/ x^2-0/ i*n*x/ 0.5*x/ (x^2-0.5*x)*sin(2*pi*n*x)

Integral de (x^2-0.5*x)*sin(2*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  - -|*sin(2*pi*n*x) dx
 |  \     2/                 
 |                           
/                            
0
∫012(x2−x2)sin⁡(x2πn) dx\int\limits_{0}^{\frac{1}{2}} \left(x^{2} - \frac{x}{2}\right) \sin{\left(x 2 \pi n \right)}\, dx 
Integral((x^2 - x/2)*sin(((2*pi)*n)*x), (x, 0, 1/2))
Respuesta (Indefinida) [src] 
                                   /                 0                   for n = 0                                                                                                                                        
                                   |                                                                                                                                                                                      
                                   | //sin(2*pi*n*x)                 \                                                                                                                                                    
                                   | ||-------------  for 2*pi*n != 0|                 //                       0                          for n = 0\                                                                     
                                   <-|<    2*pi*n                    |                 ||                                                           |                                        //       0         for n = 0\
                                   | ||                              |                 || //cos(2*pi*n*x)   x*sin(2*pi*n*x)            \            |                                        ||                          |
  /                                | \\      x           otherwise   /                 || ||------------- + ---------------  for n != 0|            |                                      x*|<-cos(2*pi*n*x)            |
 |                                 |-----------------------------------  otherwise     || ||       2  2          2*pi*n                |            |      //       0         for n = 0\     ||---------------  otherwise|
 | / 2   x\                        \               2*pi*n                              || ||   4*pi *n                                 |            |    2 ||                          |     \\     2*pi*n               /
 | |x  - -|*sin(2*pi*n*x) dx = C + ----------------------------------------------- - 2*|<-|<                                           |            | + x *|<-cos(2*pi*n*x)            | - -------------------------------
 | \     2/                                               2                            || ||               2                           |            |      ||---------------  otherwise|                  2               
 |                                                                                     || ||              x                            |            |      \\     2*pi*n               /                                  
/                                                                                      || ||              --                 otherwise |            |                                                                     
                                                                                       || \\              2                            /            |                                                                     
                                                                                       ||------------------------------------------------  otherwise|                                                                     
                                                                                       \\                     2*pi*n                                /
∫(x2−x2)sin⁡(x2πn) dx=C+x2({0for n=0−cos⁡(2πnx)2πnotherwise)−x({0for n=0−cos⁡(2πnx)2πnotherwise)2+{0for n=0−{sin⁡(2πnx)2πnfor 2πn≠0xotherwise2πnotherwise2−2({0for n=0−{xsin⁡(2πnx)2πn+cos⁡(2πnx)4π2n2for n≠0x22otherwise2πnotherwise)\int \left(x^{2} - \frac{x}{2}\right) \sin{\left(x 2 \pi n \right)}\, dx = C + x^{2} \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\cos{\left(2 \pi n x \right)}}{2 \pi n} & \text{otherwise} \end{cases}\right) - \frac{x \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\cos{\left(2 \pi n x \right)}}{2 \pi n} & \text{otherwise} \end{cases}\right)}{2} + \frac{\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\begin{cases} \frac{\sin{\left(2 \pi n x \right)}}{2 \pi n} & \text{for}\: 2 \pi n \neq 0 \\x & \text{otherwise} \end{cases}}{2 \pi n} & \text{otherwise} \end{cases}}{2} - 2 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\begin{cases} \frac{x \sin{\left(2 \pi n x \right)}}{2 \pi n} + \frac{\cos{\left(2 \pi n x \right)}}{4 \pi^{2} n^{2}} & \text{for}\: n \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases}}{2 \pi n} & \text{otherwise} \end{cases}\right)
Simplificar
Respuesta [src] 
/     1       cos(pi*n)   sin(pi*n)                                  
|- -------- + --------- + ---------  for And(n > -oo, n < oo, n != 0)
|      3  3        3  3        2  2                                  
<  4*pi *n     4*pi *n     8*pi *n                                   
|                                                                    
|                0                              otherwise            
\
{sin⁡(πn)8π2n2+cos⁡(πn)4π3n3−14π3n3for n>−∞∧n<∞∧n≠00otherwise\begin{cases} \frac{\sin{\left(\pi n \right)}}{8 \pi^{2} n^{2}} + \frac{\cos{\left(\pi n \right)}}{4 \pi^{3} n^{3}} - \frac{1}{4 \pi^{3} n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases} 
=
= 
/     1       cos(pi*n)   sin(pi*n)                                  
|- -------- + --------- + ---------  for And(n > -oo, n < oo, n != 0)
|      3  3        3  3        2  2                                  
<  4*pi *n     4*pi *n     8*pi *n                                   
|                                                                    
|                0                              otherwise            
\
{sin⁡(πn)8π2n2+cos⁡(πn)4π3n3−14π3n3for n>−∞∧n<∞∧n≠00otherwise\begin{cases} \frac{\sin{\left(\pi n \right)}}{8 \pi^{2} n^{2}} + \frac{\cos{\left(\pi n \right)}}{4 \pi^{3} n^{3}} - \frac{1}{4 \pi^{3} n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases} 
Piecewise((-1/(4*pi^3*n^3) + cos(pi*n)/(4*pi^3*n^3) + sin(pi*n)/(8*pi^2*n^2), (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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