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

Integral de sin(pi*x)*sin(pi*n*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                         
  /                         
 |                          
 |  sin(pi*x)*sin(pi*n*x) dx
 |                          
/                           
0
∫01sin⁡(πx)sin⁡(xπn) dx\int\limits_{0}^{1} \sin{\left(\pi x \right)} \sin{\left(x \pi n \right)}\, dx 
Integral(sin(pi*x)*sin((pi*n)*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                  /                           /  pi*x   pi*n*x\                                     /                          /pi*x   pi*n*x\                                 
                                  |                      2*tan|- ---- + ------|                                     |                     2*tan|---- + ------|                                 
                                  |                           \   2       2   /                                     |                          \ 2       2   /                                 
                                  |------------------------------------------------------------------  for n != 1   |-------------------------------------------------------------  for n != -1
                                  <                   2/  pi*x   pi*n*x\           2/  pi*x   pi*n*x\               <                  2/pi*x   pi*n*x\           2/pi*x   pi*n*x\             
                                  |-pi + pi*n - pi*tan |- ---- + ------| + pi*n*tan |- ---- + ------|               |pi + pi*n + pi*tan |---- + ------| + pi*n*tan |---- + ------|             
                                  |                    \   2       2   /            \   2       2   /               |                   \ 2       2   /            \ 2       2   /             
  /                               |                                                                                 |                                                                          
 |                                \                                x                                   otherwise    \                              x                                 otherwise 
 | sin(pi*x)*sin(pi*n*x) dx = C + ------------------------------------------------------------------------------- - ---------------------------------------------------------------------------
 |                                                                       2                                                                               2                                     
/
∫sin⁡(πx)sin⁡(xπn) dx=C+{2tan⁡(πnx2−πx2)πntan⁡2(πnx2−πx2)+πn−πtan⁡2(πnx2−πx2)−πfor n≠1xotherwise2−{2tan⁡(πnx2+πx2)πntan⁡2(πnx2+πx2)+πn+πtan⁡2(πnx2+πx2)+πfor n≠−1xotherwise2\int \sin{\left(\pi x \right)} \sin{\left(x \pi n \right)}\, dx = C + \frac{\begin{cases} \frac{2 \tan{\left(\frac{\pi n x}{2} - \frac{\pi x}{2} \right)}}{\pi n \tan^{2}{\left(\frac{\pi n x}{2} - \frac{\pi x}{2} \right)} + \pi n - \pi \tan^{2}{\left(\frac{\pi n x}{2} - \frac{\pi x}{2} \right)} - \pi} & \text{for}\: n \neq 1 \\x & \text{otherwise} \end{cases}}{2} - \frac{\begin{cases} \frac{2 \tan{\left(\frac{\pi n x}{2} + \frac{\pi x}{2} \right)}}{\pi n \tan^{2}{\left(\frac{\pi n x}{2} + \frac{\pi x}{2} \right)} + \pi n + \pi \tan^{2}{\left(\frac{\pi n x}{2} + \frac{\pi x}{2} \right)} + \pi} & \text{for}\: n \neq -1 \\x & \text{otherwise} \end{cases}}{2}
Simplificar
Respuesta [src] 
/   -1/2      for n = -1
|                       
|    1/2      for n = 1 
|                       
<-sin(pi*n)             
|-----------  otherwise 
|          2            
|-pi + pi*n             
\
{−12for n=−112for n=1−sin⁡(πn)πn2−πotherwise\begin{cases} - \frac{1}{2} & \text{for}\: n = -1 \\\frac{1}{2} & \text{for}\: n = 1 \\- \frac{\sin{\left(\pi n \right)}}{\pi n^{2} - \pi} & \text{otherwise} \end{cases} 
=
= 
/   -1/2      for n = -1
|                       
|    1/2      for n = 1 
|                       
<-sin(pi*n)             
|-----------  otherwise 
|          2            
|-pi + pi*n             
\
{−12for n=−112for n=1−sin⁡(πn)πn2−πotherwise\begin{cases} - \frac{1}{2} & \text{for}\: n = -1 \\\frac{1}{2} & \text{for}\: n = 1 \\- \frac{\sin{\left(\pi n \right)}}{\pi n^{2} - \pi} & \text{otherwise} \end{cases} 
Piecewise((-1/2, n = -1), (1/2, n = 1), (-sin(pi*n)/(-pi + pi*n^2), True))

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

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