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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                   
  /                   
 |                    
 |  sin(x)*sin(n*x) dx
 |                    
/                     
0
$$\int\limits_{0}^{1} \sin{\left(x \right)} \sin{\left(n x \right)}\, dx$$ 
Integral(sin(x)*sin(n*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
                            //                     2           2               \
                            ||cos(x)*sin(x)   x*cos (x)   x*sin (x)            |
                            ||------------- - --------- - ---------  for n = -1|
                            ||      2             2           2                |
                            ||                                                 |
  /                         ||     2           2                               |
 |                          ||x*cos (x)   x*sin (x)   cos(x)*sin(x)            |
 | sin(x)*sin(n*x) dx = C + |<--------- + --------- - -------------  for n = 1 |
 |                          ||    2           2             2                  |
/                           ||                                                 |
                            || cos(x)*sin(n*x)   n*cos(n*x)*sin(x)             |
                            || --------------- - -----------------   otherwise |
                            ||           2                  2                  |
                            ||     -1 + n             -1 + n                   |
                            \\                                                 /
$$\int \sin{\left(x \right)} \sin{\left(n x \right)}\, dx = C + \begin{cases} - \frac{x \sin^{2}{\left(x \right)}}{2} - \frac{x \cos^{2}{\left(x \right)}}{2} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2} & \text{for}\: n = -1 \\\frac{x \sin^{2}{\left(x \right)}}{2} + \frac{x \cos^{2}{\left(x \right)}}{2} - \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2} & \text{for}\: n = 1 \\- \frac{n \sin{\left(x \right)} \cos{\left(n x \right)}}{n^{2} - 1} + \frac{\sin{\left(n x \right)} \cos{\left(x \right)}}{n^{2} - 1} & \text{otherwise} \end{cases}$$
Simplificar
Respuesta [src] 
/     2         2                               
|  cos (1)   sin (1)   cos(1)*sin(1)            
|- ------- - ------- + -------------  for n = -1
|     2         2            2                  
|                                               
|    2         2                                
| cos (1)   sin (1)   cos(1)*sin(1)             
< ------- + ------- - -------------   for n = 1 
|    2         2            2                   
|                                               
|  cos(1)*sin(n)   n*cos(n)*sin(1)              
|  ------------- - ---------------    otherwise 
|           2                2                  
|     -1 + n           -1 + n                   
\
$$\begin{cases} - \frac{\sin^{2}{\left(1 \right)}}{2} - \frac{\cos^{2}{\left(1 \right)}}{2} + \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} & \text{for}\: n = -1 \\- \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{\cos^{2}{\left(1 \right)}}{2} + \frac{\sin^{2}{\left(1 \right)}}{2} & \text{for}\: n = 1 \\- \frac{n \sin{\left(1 \right)} \cos{\left(n \right)}}{n^{2} - 1} + \frac{\sin{\left(n \right)} \cos{\left(1 \right)}}{n^{2} - 1} & \text{otherwise} \end{cases}$$ 
=
= 
/     2         2                               
|  cos (1)   sin (1)   cos(1)*sin(1)            
|- ------- - ------- + -------------  for n = -1
|     2         2            2                  
|                                               
|    2         2                                
| cos (1)   sin (1)   cos(1)*sin(1)             
< ------- + ------- - -------------   for n = 1 
|    2         2            2                   
|                                               
|  cos(1)*sin(n)   n*cos(n)*sin(1)              
|  ------------- - ---------------    otherwise 
|           2                2                  
|     -1 + n           -1 + n                   
\
$$\begin{cases} - \frac{\sin^{2}{\left(1 \right)}}{2} - \frac{\cos^{2}{\left(1 \right)}}{2} + \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} & \text{for}\: n = -1 \\- \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{\cos^{2}{\left(1 \right)}}{2} + \frac{\sin^{2}{\left(1 \right)}}{2} & \text{for}\: n = 1 \\- \frac{n \sin{\left(1 \right)} \cos{\left(n \right)}}{n^{2} - 1} + \frac{\sin{\left(n \right)} \cos{\left(1 \right)}}{n^{2} - 1} & \text{otherwise} \end{cases}$$ 
Piecewise((-cos(1)^2/2 - sin(1)^2/2 + cos(1)*sin(1)/2, n = -1), (cos(1)^2/2 + sin(1)^2/2 - cos(1)*sin(1)/2, n = 1), (cos(1)*sin(n)/(-1 + n^2) - n*cos(n)*sin(1)/(-1 + n^2), True))

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

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