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

Integral de x*(2/L)*sin^2((pi*n*x)/L) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  l                    
  /                    
 |                     
 |    2    2/pi*n*x\   
 |  x*-*sin |------| dx
 |    l     \  l   /   
 |                     
/                      
0
∫0l2lxsin⁡2(xπnl) dx\int\limits_{0}^{l} \frac{2}{l} x \sin^{2}{\left(\frac{x \pi n}{l} \right)}\, dx 
Integral((x*(2/l))*sin(((pi*n)*x)/l)^2, (x, 0, l))
Respuesta (Indefinida) [src] 
                               /  / 2    /2*pi*n*x\          /2*pi*n*x\                 \
                               |  |l *cos|--------|   l*x*sin|--------|                 |
                               |  |      \   l    /          \   l    /                 |
                               |  |---------------- + -----------------  for n != 0     |
                               |  |        2  2             2*pi*n                      |
                               |  <    4*pi *n                                          |
                               |  |                                                     |
                               |  |                  2                                  |
                               |  |                 x                                   |
                               |  |                 --                   otherwise     2|
  /                            |  \                 2                                 x |
 |                           2*|- ------------------------------------------------- + --|
 |   2    2/pi*n*x\            \                          2                           4 /
 | x*-*sin |------| dx = C + ------------------------------------------------------------
 |   l     \  l   /                                       l                              
 |                                                                                       
/
∫2lxsin⁡2(xπnl) dx=C+2(x24−{l2cos⁡(2πnxl)4π2n2+lxsin⁡(2πnxl)2πnfor n≠0x22otherwise2)l\int \frac{2}{l} x \sin^{2}{\left(\frac{x \pi n}{l} \right)}\, dx = C + \frac{2 \left(\frac{x^{2}}{4} - \frac{\begin{cases} \frac{l^{2} \cos{\left(\frac{2 \pi n x}{l} \right)}}{4 \pi^{2} n^{2}} + \frac{l x \sin{\left(\frac{2 \pi n x}{l} \right)}}{2 \pi n} & \text{for}\: n \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases}}{2}\right)}{l}
Simplificar
Respuesta [src] 
/  / 2    2          2    2          2    2          2                    \                                  
|  |l *cos (pi*n)   l *sin (pi*n)   l *sin (pi*n)   l *cos(pi*n)*sin(pi*n)|                                  
|2*|------------- + ------------- + ------------- - ----------------------|                                  
|  |      4               4                2  2             2*pi*n        |                                  
<  \                                   4*pi *n                            /                                  
|--------------------------------------------------------------------------  for And(n > -oo, n < oo, n != 0)
|                                    l                                                                       
|                                                                                                            
\                                    0                                                  otherwise
{2(l2sin⁡2(πn)4+l2cos⁡2(πn)4−l2sin⁡(πn)cos⁡(πn)2πn+l2sin⁡2(πn)4π2n2)lfor n>−∞∧n<∞∧n≠00otherwise\begin{cases} \frac{2 \left(\frac{l^{2} \sin^{2}{\left(\pi n \right)}}{4} + \frac{l^{2} \cos^{2}{\left(\pi n \right)}}{4} - \frac{l^{2} \sin{\left(\pi n \right)} \cos{\left(\pi n \right)}}{2 \pi n} + \frac{l^{2} \sin^{2}{\left(\pi n \right)}}{4 \pi^{2} n^{2}}\right)}{l} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases} 
=
= 
/  / 2    2          2    2          2    2          2                    \                                  
|  |l *cos (pi*n)   l *sin (pi*n)   l *sin (pi*n)   l *cos(pi*n)*sin(pi*n)|                                  
|2*|------------- + ------------- + ------------- - ----------------------|                                  
|  |      4               4                2  2             2*pi*n        |                                  
<  \                                   4*pi *n                            /                                  
|--------------------------------------------------------------------------  for And(n > -oo, n < oo, n != 0)
|                                    l                                                                       
|                                                                                                            
\                                    0                                                  otherwise
{2(l2sin⁡2(πn)4+l2cos⁡2(πn)4−l2sin⁡(πn)cos⁡(πn)2πn+l2sin⁡2(πn)4π2n2)lfor n>−∞∧n<∞∧n≠00otherwise\begin{cases} \frac{2 \left(\frac{l^{2} \sin^{2}{\left(\pi n \right)}}{4} + \frac{l^{2} \cos^{2}{\left(\pi n \right)}}{4} - \frac{l^{2} \sin{\left(\pi n \right)} \cos{\left(\pi n \right)}}{2 \pi n} + \frac{l^{2} \sin^{2}{\left(\pi n \right)}}{4 \pi^{2} n^{2}}\right)}{l} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases} 
Piecewise((2*(l^2*cos(pi*n)^2/4 + l^2*sin(pi*n)^2/4 + l^2*sin(pi*n)^2/(4*pi^2*n^2) - l^2*cos(pi*n)*sin(pi*n)/(2*pi*n))/l, (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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