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Simplificación de expresiones/ Descomposición en fracciones simples/ sin((n+1)*x)/(2*(1+n))+sin((1-n)*x)/(2*(1-n))

¿Cómo vas a descomponer esta sin((n+1)*x)/(2*(1+n))+sin((1-n)*x)/(2*(1-n)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
sin((n + 1)*x)   sin((1 - n)*x)
-------------- + --------------
  2*(1 + n)        2*(1 - n)
sin⁡(x(n+1))2(n+1)+sin⁡(x(1−n))2(1−n)\frac{\sin{\left(x \left(n + 1\right) \right)}}{2 \left(n + 1\right)} + \frac{\sin{\left(x \left(1 - n\right) \right)}}{2 \left(1 - n\right)} 
sin((n + 1)*x)/((2*(1 + n))) + sin((1 - n)*x)/((2*(1 - n)))
Simplificación general [src] 
(1 + n)*sin(x*(-1 + n)) + (-1 + n)*sin(x*(1 + n))
-------------------------------------------------
                2*(1 + n)*(-1 + n)
(n−1)sin⁡(x(n+1))+(n+1)sin⁡(x(n−1))2(n−1)(n+1)\frac{\left(n - 1\right) \sin{\left(x \left(n + 1\right) \right)} + \left(n + 1\right) \sin{\left(x \left(n - 1\right) \right)}}{2 \left(n - 1\right) \left(n + 1\right)} 
((1 + n)*sin(x*(-1 + n)) + (-1 + n)*sin(x*(1 + n)))/(2*(1 + n)*(-1 + n))
Respuesta numérica [src] 
sin((n + 1)*x)/(2.0 + 2.0*n) + sin((1 - n)*x)/(2.0 - 2.0*n)
sin((n + 1)*x)/(2.0 + 2.0*n) + sin((1 - n)*x)/(2.0 - 2.0*n)
Combinatoria [src] 
-sin(x + n*x) + n*sin(x + n*x) + n*sin(-x + n*x) + sin(-x + n*x)
----------------------------------------------------------------
                       2*(1 + n)*(-1 + n)
nsin⁡(nx−x)+nsin⁡(nx+x)+sin⁡(nx−x)−sin⁡(nx+x)2(n−1)(n+1)\frac{n \sin{\left(n x - x \right)} + n \sin{\left(n x + x \right)} + \sin{\left(n x - x \right)} - \sin{\left(n x + x \right)}}{2 \left(n - 1\right) \left(n + 1\right)} 
(-sin(x + n*x) + n*sin(x + n*x) + n*sin(-x + n*x) + sin(-x + n*x))/(2*(1 + n)*(-1 + n))
Denominador racional [src] 
-2*sin(x + n*x) + 2*sin(-x + n*x) + 2*n*sin(x + n*x) + 2*n*sin(-x + n*x)
------------------------------------------------------------------------
                          (-2 + 2*n)*(2 + 2*n)
2nsin⁡(nx−x)+2nsin⁡(nx+x)+2sin⁡(nx−x)−2sin⁡(nx+x)(2n−2)(2n+2)\frac{2 n \sin{\left(n x - x \right)} + 2 n \sin{\left(n x + x \right)} + 2 \sin{\left(n x - x \right)} - 2 \sin{\left(n x + x \right)}}{\left(2 n - 2\right) \left(2 n + 2\right)} 
(-2*sin(x + n*x) + 2*sin(-x + n*x) + 2*n*sin(x + n*x) + 2*n*sin(-x + n*x))/((-2 + 2*n)*(2 + 2*n))
Unión de expresiones racionales [src] 
(1 + n)*sin(x*(1 - n)) + (1 - n)*sin(x*(1 + n))
-----------------------------------------------
               2*(1 + n)*(1 - n)
(1−n)sin⁡(x(n+1))+(n+1)sin⁡(x(1−n))2(1−n)(n+1)\frac{\left(1 - n\right) \sin{\left(x \left(n + 1\right) \right)} + \left(n + 1\right) \sin{\left(x \left(1 - n\right) \right)}}{2 \left(1 - n\right) \left(n + 1\right)} 
((1 + n)*sin(x*(1 - n)) + (1 - n)*sin(x*(1 + n)))/(2*(1 + n)*(1 - n))
Abrimos la expresión [src] 
cos(n*x)*sin(x)   cos(x)*sin(n*x)   cos(n*x)*sin(x)   cos(x)*sin(n*x)
--------------- + --------------- + --------------- - ---------------
    2 - 2*n           2 + 2*n           2 + 2*n           2 - 2*n
sin⁡(x)cos⁡(nx)2n+2+sin⁡(nx)cos⁡(x)2n+2+sin⁡(x)cos⁡(nx)2−2n−sin⁡(nx)cos⁡(x)2−2n\frac{\sin{\left(x \right)} \cos{\left(n x \right)}}{2 n + 2} + \frac{\sin{\left(n x \right)} \cos{\left(x \right)}}{2 n + 2} + \frac{\sin{\left(x \right)} \cos{\left(n x \right)}}{2 - 2 n} - \frac{\sin{\left(n x \right)} \cos{\left(x \right)}}{2 - 2 n} 
cos(n*x)*sin(x)/(2 - 2*n) + cos(x)*sin(n*x)/(2 + 2*n) + cos(n*x)*sin(x)/(2 + 2*n) - cos(x)*sin(n*x)/(2 - 2*n)
Compilar la expresión [src] 
sin((1 - n)*x)   sin((n + 1)*x)
-------------- + --------------
   2 - 2*n          2 + 2*n
sin⁡(x(n+1))2n+2+sin⁡(x(1−n))2−2n\frac{\sin{\left(x \left(n + 1\right) \right)}}{2 n + 2} + \frac{\sin{\left(x \left(1 - n\right) \right)}}{2 - 2 n} 
sin((1 - n)*x)/(2 - 2*n) + sin((n + 1)*x)/(2 + 2*n)
Denominador común [src] 
-sin(x + n*x) + n*sin(x + n*x) + n*sin(-x + n*x) + sin(-x + n*x)
----------------------------------------------------------------
                                   2                            
                           -2 + 2*n
nsin⁡(nx−x)+nsin⁡(nx+x)+sin⁡(nx−x)−sin⁡(nx+x)2n2−2\frac{n \sin{\left(n x - x \right)} + n \sin{\left(n x + x \right)} + \sin{\left(n x - x \right)} - \sin{\left(n x + x \right)}}{2 n^{2} - 2} 
(-sin(x + n*x) + n*sin(x + n*x) + n*sin(-x + n*x) + sin(-x + n*x))/(-2 + 2*n^2)
Parte trigonométrica [src] 
               1                                 1               
------------------------------- + -------------------------------
             /  pi            \                /  pi            \
(2 - 2*n)*sec|- -- + x*(1 - n)|   (2 + 2*n)*sec|- -- + x*(1 + n)|
             \  2             /                \  2             /
1(2n+2)sec⁡(x(n+1)−π2)+1(2−2n)sec⁡(x(1−n)−π2)\frac{1}{\left(2 n + 2\right) \sec{\left(x \left(n + 1\right) - \frac{\pi}{2} \right)}} + \frac{1}{\left(2 - 2 n\right) \sec{\left(x \left(1 - n\right) - \frac{\pi}{2} \right)}} 
             /x*(1 + n)\                       /x*(1 - n)\       
        2*cot|---------|                  2*cot|---------|       
             \    2    /                       \    2    /       
------------------------------- + -------------------------------
/       2/x*(1 + n)\\             /       2/x*(1 - n)\\          
|1 + cot |---------||*(2 + 2*n)   |1 + cot |---------||*(2 - 2*n)
\        \    2    //             \        \    2    //
2cot⁡(x(n+1)2)(2n+2)(cot⁡2(x(n+1)2)+1)+2cot⁡(x(1−n)2)(2−2n)(cot⁡2(x(1−n)2)+1)\frac{2 \cot{\left(\frac{x \left(n + 1\right)}{2} \right)}}{\left(2 n + 2\right) \left(\cot^{2}{\left(\frac{x \left(n + 1\right)}{2} \right)} + 1\right)} + \frac{2 \cot{\left(\frac{x \left(1 - n\right)}{2} \right)}}{\left(2 - 2 n\right) \left(\cot^{2}{\left(\frac{x \left(1 - n\right)}{2} \right)} + 1\right)} 
             /x*(1 + n)\                       /x*(1 - n)\       
        2*tan|---------|                  2*tan|---------|       
             \    2    /                       \    2    /       
------------------------------- + -------------------------------
/       2/x*(1 + n)\\             /       2/x*(1 - n)\\          
|1 + tan |---------||*(2 + 2*n)   |1 + tan |---------||*(2 - 2*n)
\        \    2    //             \        \    2    //
2tan⁡(x(n+1)2)(2n+2)(tan⁡2(x(n+1)2)+1)+2tan⁡(x(1−n)2)(2−2n)(tan⁡2(x(1−n)2)+1)\frac{2 \tan{\left(\frac{x \left(n + 1\right)}{2} \right)}}{\left(2 n + 2\right) \left(\tan^{2}{\left(\frac{x \left(n + 1\right)}{2} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{x \left(1 - n\right)}{2} \right)}}{\left(2 - 2 n\right) \left(\tan^{2}{\left(\frac{x \left(1 - n\right)}{2} \right)} + 1\right)} 
           1                          1            
------------------------ + ------------------------
(2 - 2*n)*csc(x*(1 - n))   (2 + 2*n)*csc(x*(1 + n))
1(2n+2)csc⁡(x(n+1))+1(2−2n)csc⁡(x(1−n))\frac{1}{\left(2 n + 2\right) \csc{\left(x \left(n + 1\right) \right)}} + \frac{1}{\left(2 - 2 n\right) \csc{\left(x \left(1 - n\right) \right)}} 
             1                               1                
--------------------------- - --------------------------------
             /    pi      \                /  pi             \
(2 + 2*n)*sec|x - -- + n*x|   (2 - 2*n)*sec|- -- + x*(-1 + n)|
             \    2       /                \  2              /
1(2n+2)sec⁡(nx+x−π2)−1(2−2n)sec⁡(x(n−1)−π2)\frac{1}{\left(2 n + 2\right) \sec{\left(n x + x - \frac{\pi}{2} \right)}} - \frac{1}{\left(2 - 2 n\right) \sec{\left(x \left(n - 1\right) - \frac{\pi}{2} \right)}} 
    1                          1                   
---------*sin((n + 1)*x) + ---------*sin((1 - n)*x)
2*(1 + n)                  2*(1 - n)
12(1−n)sin⁡(x(1−n))+12(n+1)sin⁡(x(n+1))\frac{1}{2 \left(1 - n\right)} \sin{\left(x \left(1 - n\right) \right)} + \frac{1}{2 \left(n + 1\right)} \sin{\left(x \left(n + 1\right) \right)} 
   /    pi      \      /  pi             \
cos|x - -- + n*x|   cos|- -- + x*(-1 + n)|
   \    2       /      \  2              /
----------------- - ----------------------
     2 + 2*n               2 - 2*n
cos⁡(nx+x−π2)2n+2−cos⁡(x(n−1)−π2)2−2n\frac{\cos{\left(n x + x - \frac{\pi}{2} \right)}}{2 n + 2} - \frac{\cos{\left(x \left(n - 1\right) - \frac{\pi}{2} \right)}}{2 - 2 n} 
              /x*(-1 + n)\                        /x   n*x\       
         2*cot|----------|                   2*cot|- + ---|       
              \    2     /                        \2    2 /       
- -------------------------------- + -----------------------------
  /       2/x*(-1 + n)\\             /       2/x   n*x\\          
  |1 + cot |----------||*(2 - 2*n)   |1 + cot |- + ---||*(2 + 2*n)
  \        \    2     //             \        \2    2 //
2cot⁡(nx2+x2)(2n+2)(cot⁡2(nx2+x2)+1)−2cot⁡(x(n−1)2)(2−2n)(cot⁡2(x(n−1)2)+1)\frac{2 \cot{\left(\frac{n x}{2} + \frac{x}{2} \right)}}{\left(2 n + 2\right) \left(\cot^{2}{\left(\frac{n x}{2} + \frac{x}{2} \right)} + 1\right)} - \frac{2 \cot{\left(\frac{x \left(n - 1\right)}{2} \right)}}{\left(2 - 2 n\right) \left(\cot^{2}{\left(\frac{x \left(n - 1\right)}{2} \right)} + 1\right)} 
                                 /x   n*x\       
                            2*tan|- + ---|       
  sin(x*(-1 + n))                \2    2 /       
- --------------- + -----------------------------
      2 - 2*n       /       2/x   n*x\\          
                    |1 + tan |- + ---||*(2 + 2*n)
                    \        \2    2 //
2tan⁡(nx2+x2)(2n+2)(tan⁡2(nx2+x2)+1)−sin⁡(x(n−1))2−2n\frac{2 \tan{\left(\frac{n x}{2} + \frac{x}{2} \right)}}{\left(2 n + 2\right) \left(\tan^{2}{\left(\frac{n x}{2} + \frac{x}{2} \right)} + 1\right)} - \frac{\sin{\left(x \left(n - 1\right) \right)}}{2 - 2 n} 
              /x*(-1 + n)\                        /x   n*x\       
         2*tan|----------|                   2*tan|- + ---|       
              \    2     /                        \2    2 /       
- -------------------------------- + -----------------------------
  /       2/x*(-1 + n)\\             /       2/x   n*x\\          
  |1 + tan |----------||*(2 - 2*n)   |1 + tan |- + ---||*(2 + 2*n)
  \        \    2     //             \        \2    2 //
2tan⁡(nx2+x2)(2n+2)(tan⁡2(nx2+x2)+1)−2tan⁡(x(n−1)2)(2−2n)(tan⁡2(x(n−1)2)+1)\frac{2 \tan{\left(\frac{n x}{2} + \frac{x}{2} \right)}}{\left(2 n + 2\right) \left(\tan^{2}{\left(\frac{n x}{2} + \frac{x}{2} \right)} + 1\right)} - \frac{2 \tan{\left(\frac{x \left(n - 1\right)}{2} \right)}}{\left(2 - 2 n\right) \left(\tan^{2}{\left(\frac{x \left(n - 1\right)}{2} \right)} + 1\right)} 
sin(x + n*x)   sin(x*(-1 + n))
------------ - ---------------
  2 + 2*n          2 - 2*n
sin⁡(nx+x)2n+2−sin⁡(x(n−1))2−2n\frac{\sin{\left(n x + x \right)}}{2 n + 2} - \frac{\sin{\left(x \left(n - 1\right) \right)}}{2 - 2 n} 
   /  pi            \      /  pi            \
cos|- -- + x*(1 - n)|   cos|- -- + x*(1 + n)|
   \  2             /      \  2             /
--------------------- + ---------------------
       2 - 2*n                 2 + 2*n
cos⁡(x(n+1)−π2)2n+2+cos⁡(x(1−n)−π2)2−2n\frac{\cos{\left(x \left(n + 1\right) - \frac{\pi}{2} \right)}}{2 n + 2} + \frac{\cos{\left(x \left(1 - n\right) - \frac{\pi}{2} \right)}}{2 - 2 n} 
          1                          1            
---------------------- - -------------------------
(2 + 2*n)*csc(x + n*x)   (2 - 2*n)*csc(x*(-1 + n))
1(2n+2)csc⁡(nx+x)−1(2−2n)csc⁡(x(n−1))\frac{1}{\left(2 n + 2\right) \csc{\left(n x + x \right)}} - \frac{1}{\left(2 - 2 n\right) \csc{\left(x \left(n - 1\right) \right)}} 
sin(x*(1 - n))   sin(x*(1 + n))
-------------- + --------------
   2 - 2*n          2 + 2*n
sin⁡(x(n+1))2n+2+sin⁡(x(1−n))2−2n\frac{\sin{\left(x \left(n + 1\right) \right)}}{2 n + 2} + \frac{\sin{\left(x \left(1 - n\right) \right)}}{2 - 2 n} 
sin(x*(1 - n))/(2 - 2*n) + sin(x*(1 + n))/(2 + 2*n)
Potencias [src] 
    /   -I*x*(1 - n)    I*x*(1 - n)\     /   -I*x*(1 + n)    I*x*(1 + n)\
  I*\- e             + e           /   I*\- e             + e           /
- ---------------------------------- - ----------------------------------
             2*(2 - 2*n)                          2*(2 + 2*n)
−i(eix(n+1)−e−ix(n+1))2(2n+2)−i(eix(1−n)−e−ix(1−n))2(2−2n)- \frac{i \left(e^{i x \left(n + 1\right)} - e^{- i x \left(n + 1\right)}\right)}{2 \left(2 n + 2\right)} - \frac{i \left(e^{i x \left(1 - n\right)} - e^{- i x \left(1 - n\right)}\right)}{2 \left(2 - 2 n\right)} 
sin(x*(1 - n))   sin(x*(1 + n))
-------------- + --------------
   2 - 2*n          2 + 2*n
sin⁡(x(n+1))2n+2+sin⁡(x(1−n))2−2n\frac{\sin{\left(x \left(n + 1\right) \right)}}{2 n + 2} + \frac{\sin{\left(x \left(1 - n\right) \right)}}{2 - 2 n} 
sin(x*(1 - n))/(2 - 2*n) + sin(x*(1 + n))/(2 + 2*n)
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