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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
sin((1 - m)*x)   sin((m + 1)*x)
-------------- - --------------
  2*(1 - m)        2*(1 + m)
$$- \frac{\sin{\left(x \left(m + 1\right) \right)}}{2 \left(m + 1\right)} + \frac{\sin{\left(x \left(1 - m\right) \right)}}{2 \left(1 - m\right)}$$ 
sin((1 - m)*x)/((2*(1 - m))) - sin((m + 1)*x)/(2*(1 + m))
Simplificación general [src] 
(1 + m)*sin(x*(-1 + m)) + (1 - m)*sin(x*(1 + m))
------------------------------------------------
               2*(1 + m)*(-1 + m)
$$\frac{\left(1 - m\right) \sin{\left(x \left(m + 1\right) \right)} + \left(m + 1\right) \sin{\left(x \left(m - 1\right) \right)}}{2 \left(m - 1\right) \left(m + 1\right)}$$ 
((1 + m)*sin(x*(-1 + m)) + (1 - m)*sin(x*(1 + m)))/(2*(1 + m)*(-1 + m))
Respuesta numérica [src] 
sin((1 - m)*x)/(2.0 - 2.0*m) - sin((m + 1)*x)/(2.0 + 2.0*m)
sin((1 - m)*x)/(2.0 - 2.0*m) - sin((m + 1)*x)/(2.0 + 2.0*m)
Denominador racional [src] 
2*sin(x + m*x) + 2*sin(-x + m*x) - 2*m*sin(x + m*x) + 2*m*sin(-x + m*x)
-----------------------------------------------------------------------
                          (-2 + 2*m)*(2 + 2*m)
$$\frac{2 m \sin{\left(m x - x \right)} - 2 m \sin{\left(m x + x \right)} + 2 \sin{\left(m x - x \right)} + 2 \sin{\left(m x + x \right)}}{\left(2 m - 2\right) \left(2 m + 2\right)}$$ 
(2*sin(x + m*x) + 2*sin(-x + m*x) - 2*m*sin(x + m*x) + 2*m*sin(-x + m*x))/((-2 + 2*m)*(2 + 2*m))
Combinatoria [src] 
-(-sin(x + m*x) - sin(-x + m*x) + m*sin(x + m*x) - m*sin(-x + m*x)) 
--------------------------------------------------------------------
                         2*(1 + m)*(-1 + m)
$$- \frac{- m \sin{\left(m x - x \right)} + m \sin{\left(m x + x \right)} - \sin{\left(m x - x \right)} - \sin{\left(m x + x \right)}}{2 \left(m - 1\right) \left(m + 1\right)}$$ 
-(-sin(x + m*x) - sin(-x + m*x) + m*sin(x + m*x) - m*sin(-x + m*x))/(2*(1 + m)*(-1 + m))
Denominador común [src] 
-(-sin(x + m*x) - sin(-x + m*x) + m*sin(x + m*x) - m*sin(-x + m*x)) 
--------------------------------------------------------------------
                                     2                              
                             -2 + 2*m
$$- \frac{- m \sin{\left(m x - x \right)} + m \sin{\left(m x + x \right)} - \sin{\left(m x - x \right)} - \sin{\left(m x + x \right)}}{2 m^{2} - 2}$$ 
-(-sin(x + m*x) - sin(-x + m*x) + m*sin(x + m*x) - m*sin(-x + m*x))/(-2 + 2*m^2)
Potencias [src] 
  /   -I*x*(1 + m)    I*x*(1 + m)\     /   -I*x*(1 - m)    I*x*(1 - m)\
I*\- e             + e           /   I*\- e             + e           /
---------------------------------- - ----------------------------------
           2*(2 + 2*m)                          2*(2 - 2*m)
$$\frac{i \left(e^{i x \left(m + 1\right)} - e^{- i x \left(m + 1\right)}\right)}{2 \left(2 m + 2\right)} - \frac{i \left(e^{i x \left(1 - m\right)} - e^{- i x \left(1 - m\right)}\right)}{2 \left(2 - 2 m\right)}$$ 
sin(x*(1 - m))   sin(x*(1 + m))
-------------- - --------------
   2 - 2*m          2 + 2*m
$$- \frac{\sin{\left(x \left(m + 1\right) \right)}}{2 m + 2} + \frac{\sin{\left(x \left(1 - m\right) \right)}}{2 - 2 m}$$ 
sin(x*(1 - m))/(2 - 2*m) - sin(x*(1 + m))/(2 + 2*m)
Unión de expresiones racionales [src] 
(1 + m)*sin(x*(1 - m)) - (1 - m)*sin(x*(1 + m))
-----------------------------------------------
               2*(1 + m)*(1 - m)
$$\frac{- \left(1 - m\right) \sin{\left(x \left(m + 1\right) \right)} + \left(m + 1\right) \sin{\left(x \left(1 - m\right) \right)}}{2 \left(1 - m\right) \left(m + 1\right)}$$ 
((1 + m)*sin(x*(1 - m)) - (1 - m)*sin(x*(1 + m)))/(2*(1 + m)*(1 - m))
Abrimos la expresión [src] 
cos(m*x)*sin(x)   cos(x)*sin(m*x)   cos(x)*sin(m*x)   cos(m*x)*sin(x)
--------------- - --------------- - --------------- - ---------------
    2 - 2*m           2 - 2*m           2 + 2*m           2 + 2*m
$$- \frac{\sin{\left(x \right)} \cos{\left(m x \right)}}{2 m + 2} - \frac{\sin{\left(m x \right)} \cos{\left(x \right)}}{2 m + 2} + \frac{\sin{\left(x \right)} \cos{\left(m x \right)}}{2 - 2 m} - \frac{\sin{\left(m x \right)} \cos{\left(x \right)}}{2 - 2 m}$$ 
cos(m*x)*sin(x)/(2 - 2*m) - cos(x)*sin(m*x)/(2 - 2*m) - cos(x)*sin(m*x)/(2 + 2*m) - cos(m*x)*sin(x)/(2 + 2*m)
Compilar la expresión [src] 
sin((1 - m)*x)   sin((m + 1)*x)
-------------- - --------------
   2 - 2*m          2 + 2*m
$$- \frac{\sin{\left(x \left(m + 1\right) \right)}}{2 m + 2} + \frac{\sin{\left(x \left(1 - m\right) \right)}}{2 - 2 m}$$ 
sin((1 - m)*x)/(2 - 2*m) - sin((m + 1)*x)/(2 + 2*m)
Parte trigonométrica [src] 
               1                                 1               
------------------------------- - -------------------------------
             /  pi            \                /  pi            \
(2 - 2*m)*sec|- -- + x*(1 - m)|   (2 + 2*m)*sec|- -- + x*(1 + m)|
             \  2             /                \  2             /
$$- \frac{1}{\left(2 m + 2\right) \sec{\left(x \left(m + 1\right) - \frac{\pi}{2} \right)}} + \frac{1}{\left(2 - 2 m\right) \sec{\left(x \left(1 - m\right) - \frac{\pi}{2} \right)}}$$ 
              /x*(-1 + m)\                        /x   m*x\       
         2*tan|----------|                   2*tan|- + ---|       
              \    2     /                        \2    2 /       
- -------------------------------- - -----------------------------
  /       2/x*(-1 + m)\\             /       2/x   m*x\\          
  |1 + tan |----------||*(2 - 2*m)   |1 + tan |- + ---||*(2 + 2*m)
  \        \    2     //             \        \2    2 //
$$- \frac{2 \tan{\left(\frac{m x}{2} + \frac{x}{2} \right)}}{\left(2 m + 2\right) \left(\tan^{2}{\left(\frac{m x}{2} + \frac{x}{2} \right)} + 1\right)} - \frac{2 \tan{\left(\frac{x \left(m - 1\right)}{2} \right)}}{\left(2 - 2 m\right) \left(\tan^{2}{\left(\frac{x \left(m - 1\right)}{2} \right)} + 1\right)}$$ 
   /  pi            \      /  pi            \
cos|- -- + x*(1 - m)|   cos|- -- + x*(1 + m)|
   \  2             /      \  2             /
--------------------- - ---------------------
       2 - 2*m                 2 + 2*m
$$- \frac{\cos{\left(x \left(m + 1\right) - \frac{\pi}{2} \right)}}{2 m + 2} + \frac{\cos{\left(x \left(1 - m\right) - \frac{\pi}{2} \right)}}{2 - 2 m}$$ 
    1                      sin((m + 1)*x)
---------*sin((1 - m)*x) - --------------
2*(1 - m)                    2*(1 + m)
$$\frac{1}{2 \left(1 - m\right)} \sin{\left(x \left(1 - m\right) \right)} - \frac{\sin{\left(x \left(m + 1\right) \right)}}{2 \left(m + 1\right)}$$ 
  sin(x*(-1 + m))   sin(x + m*x)
- --------------- - ------------
      2 - 2*m         2 + 2*m
$$- \frac{\sin{\left(m x + x \right)}}{2 m + 2} - \frac{\sin{\left(x \left(m - 1\right) \right)}}{2 - 2 m}$$ 
               /x*(1 + m)\                       /x*(1 - m)\       
          2*cot|---------|                  2*cot|---------|       
               \    2    /                       \    2    /       
- ------------------------------- + -------------------------------
  /       2/x*(1 + m)\\             /       2/x*(1 - m)\\          
  |1 + cot |---------||*(2 + 2*m)   |1 + cot |---------||*(2 - 2*m)
  \        \    2    //             \        \    2    //
$$- \frac{2 \cot{\left(\frac{x \left(m + 1\right)}{2} \right)}}{\left(2 m + 2\right) \left(\cot^{2}{\left(\frac{x \left(m + 1\right)}{2} \right)} + 1\right)} + \frac{2 \cot{\left(\frac{x \left(1 - m\right)}{2} \right)}}{\left(2 - 2 m\right) \left(\cot^{2}{\left(\frac{x \left(1 - m\right)}{2} \right)} + 1\right)}$$ 
     /  pi             \      /    pi      \
  cos|- -- + x*(-1 + m)|   cos|x - -- + m*x|
     \  2              /      \    2       /
- ---------------------- - -----------------
         2 - 2*m                2 + 2*m
$$- \frac{\cos{\left(m x + x - \frac{\pi}{2} \right)}}{2 m + 2} - \frac{\cos{\left(x \left(m - 1\right) - \frac{\pi}{2} \right)}}{2 - 2 m}$$ 
sin(x*(1 - m))   sin(x*(1 + m))
-------------- - --------------
   2 - 2*m          2 + 2*m
$$- \frac{\sin{\left(x \left(m + 1\right) \right)}}{2 m + 2} + \frac{\sin{\left(x \left(1 - m\right) \right)}}{2 - 2 m}$$ 
                 1                                1             
- -------------------------------- - ---------------------------
               /  pi             \                /    pi      \
  (2 - 2*m)*sec|- -- + x*(-1 + m)|   (2 + 2*m)*sec|x - -- + m*x|
               \  2              /                \    2       /
$$- \frac{1}{\left(2 m + 2\right) \sec{\left(m x + x - \frac{\pi}{2} \right)}} - \frac{1}{\left(2 - 2 m\right) \sec{\left(x \left(m - 1\right) - \frac{\pi}{2} \right)}}$$ 
           1                          1            
------------------------ - ------------------------
(2 - 2*m)*csc(x*(1 - m))   (2 + 2*m)*csc(x*(1 + m))
$$- \frac{1}{\left(2 m + 2\right) \csc{\left(x \left(m + 1\right) \right)}} + \frac{1}{\left(2 - 2 m\right) \csc{\left(x \left(1 - m\right) \right)}}$$ 
              /x*(-1 + m)\                        /x   m*x\       
         2*cot|----------|                   2*cot|- + ---|       
              \    2     /                        \2    2 /       
- -------------------------------- - -----------------------------
  /       2/x*(-1 + m)\\             /       2/x   m*x\\          
  |1 + cot |----------||*(2 - 2*m)   |1 + cot |- + ---||*(2 + 2*m)
  \        \    2     //             \        \2    2 //
$$- \frac{2 \cot{\left(\frac{m x}{2} + \frac{x}{2} \right)}}{\left(2 m + 2\right) \left(\cot^{2}{\left(\frac{m x}{2} + \frac{x}{2} \right)} + 1\right)} - \frac{2 \cot{\left(\frac{x \left(m - 1\right)}{2} \right)}}{\left(2 - 2 m\right) \left(\cot^{2}{\left(\frac{x \left(m - 1\right)}{2} \right)} + 1\right)}$$ 
               /x*(1 + m)\                       /x*(1 - m)\       
          2*tan|---------|                  2*tan|---------|       
               \    2    /                       \    2    /       
- ------------------------------- + -------------------------------
  /       2/x*(1 + m)\\             /       2/x*(1 - m)\\          
  |1 + tan |---------||*(2 + 2*m)   |1 + tan |---------||*(2 - 2*m)
  \        \    2    //             \        \    2    //
$$- \frac{2 \tan{\left(\frac{x \left(m + 1\right)}{2} \right)}}{\left(2 m + 2\right) \left(\tan^{2}{\left(\frac{x \left(m + 1\right)}{2} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{x \left(1 - m\right)}{2} \right)}}{\left(2 - 2 m\right) \left(\tan^{2}{\left(\frac{x \left(1 - m\right)}{2} \right)} + 1\right)}$$ 
                                 /x   m*x\       
                            2*tan|- + ---|       
  sin(x*(-1 + m))                \2    2 /       
- --------------- - -----------------------------
      2 - 2*m       /       2/x   m*x\\          
                    |1 + tan |- + ---||*(2 + 2*m)
                    \        \2    2 //
$$- \frac{2 \tan{\left(\frac{m x}{2} + \frac{x}{2} \right)}}{\left(2 m + 2\right) \left(\tan^{2}{\left(\frac{m x}{2} + \frac{x}{2} \right)} + 1\right)} - \frac{\sin{\left(x \left(m - 1\right) \right)}}{2 - 2 m}$$ 
              1                         1           
- ------------------------- - ----------------------
  (2 - 2*m)*csc(x*(-1 + m))   (2 + 2*m)*csc(x + m*x)
$$- \frac{1}{\left(2 m + 2\right) \csc{\left(m x + x \right)}} - \frac{1}{\left(2 - 2 m\right) \csc{\left(x \left(m - 1\right) \right)}}$$ 
-1/((2 - 2*m)*csc(x*(-1 + m))) - 1/((2 + 2*m)*csc(x + m*x))
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