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

Integral de 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           
 |  x*sin (pi*n*x) dx
 |                   
/                    
0
$$\int\limits_{0}^{1} x \sin^{2}{\left(x \pi n \right)}\, dx$$ 
Integral(x*sin((pi*n)*x)^2, (x, 0, 1))
Respuesta (Indefinida) [src] 
                           /cos(2*pi*n*x)   x*sin(2*pi*n*x)                 
                           |------------- + ---------------  for n != 0     
                           |       2  2          2*pi*n                     
                           |   4*pi *n                                      
                           <                                                
                           |               2                                
  /                        |              x                                 
 |                         |              --                 otherwise     2
 |      2                  \              2                               x 
 | x*sin (pi*n*x) dx = C - -------------------------------------------- + --
 |                                              2                         4 
/
$$\int x \sin^{2}{\left(x \pi n \right)}\, dx = C + \frac{x^{2}}{4} - \frac{\begin{cases} \frac{x \sin{\left(2 \pi n x \right)}}{2 \pi n} + \frac{\cos{\left(2 \pi n x \right)}}{4 \pi^{2} n^{2}} & \text{for}\: n \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases}}{2}$$
Simplificar
Respuesta [src] 
/   2            2            2                                                              
|cos (pi*n)   sin (pi*n)   sin (pi*n)   cos(pi*n)*sin(pi*n)                                  
|---------- + ---------- + ---------- - -------------------  for And(n > -oo, n < oo, n != 0)
<    4            4             2  2           2*pi*n                                        
|                           4*pi *n                                                          
|                                                                                            
\                            0                                          otherwise
$$\begin{cases} \frac{\sin^{2}{\left(\pi n \right)}}{4} + \frac{\cos^{2}{\left(\pi n \right)}}{4} - \frac{\sin{\left(\pi n \right)} \cos{\left(\pi n \right)}}{2 \pi n} + \frac{\sin^{2}{\left(\pi n \right)}}{4 \pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$ 
=
= 
/   2            2            2                                                              
|cos (pi*n)   sin (pi*n)   sin (pi*n)   cos(pi*n)*sin(pi*n)                                  
|---------- + ---------- + ---------- - -------------------  for And(n > -oo, n < oo, n != 0)
<    4            4             2  2           2*pi*n                                        
|                           4*pi *n                                                          
|                                                                                            
\                            0                                          otherwise
$$\begin{cases} \frac{\sin^{2}{\left(\pi n \right)}}{4} + \frac{\cos^{2}{\left(\pi n \right)}}{4} - \frac{\sin{\left(\pi n \right)} \cos{\left(\pi n \right)}}{2 \pi n} + \frac{\sin^{2}{\left(\pi n \right)}}{4 \pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$ 
Piecewise((cos(pi*n)^2/4 + sin(pi*n)^2/4 + sin(pi*n)^2/(4*pi^2*n^2) - cos(pi*n)*sin(pi*n)/(2*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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