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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
sin((a + 1/2)*x)   sin((1/2 - a)*x)
---------------- + ----------------
  2*(a + 1/2)        2*(1/2 - a)
$$\frac{\sin{\left(x \left(a + \frac{1}{2}\right) \right)}}{2 \left(a + \frac{1}{2}\right)} + \frac{\sin{\left(x \left(\frac{1}{2} - a\right) \right)}}{2 \left(\frac{1}{2} - a\right)}$$ 
sin((a + 1/2)*x)/((2*(a + 1/2))) + sin((1/2 - a)*x)/((2*(1/2 - a)))
Simplificación general [src] 
(1 + 2*a)*sin(x*(-1/2 + a)) + (-1 + 2*a)*sin(x*(1/2 + a))
---------------------------------------------------------
                   (1 + 2*a)*(-1 + 2*a)
$$\frac{\left(2 a - 1\right) \sin{\left(x \left(a + \frac{1}{2}\right) \right)} + \left(2 a + 1\right) \sin{\left(x \left(a - \frac{1}{2}\right) \right)}}{\left(2 a - 1\right) \left(2 a + 1\right)}$$ 
((1 + 2*a)*sin(x*(-1/2 + a)) + (-1 + 2*a)*sin(x*(1/2 + a)))/((1 + 2*a)*(-1 + 2*a))
Respuesta numérica [src] 
sin((a + 1/2)*x)/(1.0 + 2.0*a) + sin((1/2 - a)*x)/(1.0 - 2.0*a)
sin((a + 1/2)*x)/(1.0 + 2.0*a) + sin((1/2 - a)*x)/(1.0 - 2.0*a)
Denominador racional [src] 
       /x      \        /  x      \          /x      \          /  x      \
- 4*sin|- + a*x| + 4*sin|- - + a*x| + 8*a*sin|- + a*x| + 8*a*sin|- - + a*x|
       \2      /        \  2      /          \2      /          \  2      /
---------------------------------------------------------------------------
                            (-2 + 4*a)*(2 + 4*a)
$$\frac{8 a \sin{\left(a x - \frac{x}{2} \right)} + 8 a \sin{\left(a x + \frac{x}{2} \right)} + 4 \sin{\left(a x - \frac{x}{2} \right)} - 4 \sin{\left(a x + \frac{x}{2} \right)}}{\left(4 a - 2\right) \left(4 a + 2\right)}$$ 
(-4*sin(x/2 + a*x) + 4*sin(-x/2 + a*x) + 8*a*sin(x/2 + a*x) + 8*a*sin(-x/2 + a*x))/((-2 + 4*a)*(2 + 4*a))
Denominador común [src] 
     /x      \          /x      \          /  x      \      /  x      \
- sin|- + a*x| + 2*a*sin|- + a*x| + 2*a*sin|- - + a*x| + sin|- - + a*x|
     \2      /          \2      /          \  2      /      \  2      /
-----------------------------------------------------------------------
                                       2                               
                               -1 + 4*a
$$\frac{2 a \sin{\left(a x - \frac{x}{2} \right)} + 2 a \sin{\left(a x + \frac{x}{2} \right)} + \sin{\left(a x - \frac{x}{2} \right)} - \sin{\left(a x + \frac{x}{2} \right)}}{4 a^{2} - 1}$$ 
(-sin(x/2 + a*x) + 2*a*sin(x/2 + a*x) + 2*a*sin(-x/2 + a*x) + sin(-x/2 + a*x))/(-1 + 4*a^2)
Potencias [src] 
    /   -I*x*(1/2 - a)    I*x*(1/2 - a)\     /   -I*x*(1/2 + a)    I*x*(1/2 + a)\
  I*\- e               + e             /   I*\- e               + e             /
- -------------------------------------- - --------------------------------------
               2*(1 - 2*a)                              2*(1 + 2*a)
$$- \frac{i \left(e^{i x \left(a + \frac{1}{2}\right)} - e^{- i x \left(a + \frac{1}{2}\right)}\right)}{2 \left(2 a + 1\right)} - \frac{i \left(e^{i x \left(\frac{1}{2} - a\right)} - e^{- i x \left(\frac{1}{2} - a\right)}\right)}{2 \left(1 - 2 a\right)}$$ 
sin(x*(1/2 - a))   sin(x*(1/2 + a))
---------------- + ----------------
    1 - 2*a            1 + 2*a
$$\frac{\sin{\left(x \left(a + \frac{1}{2}\right) \right)}}{2 a + 1} + \frac{\sin{\left(x \left(\frac{1}{2} - a\right) \right)}}{1 - 2 a}$$ 
sin(x*(1/2 - a))/(1 - 2*a) + sin(x*(1/2 + a))/(1 + 2*a)
Combinatoria [src] 
     /x      \          /x      \          /  x      \      /  x      \
- sin|- + a*x| + 2*a*sin|- + a*x| + 2*a*sin|- - + a*x| + sin|- - + a*x|
     \2      /          \2      /          \  2      /      \  2      /
-----------------------------------------------------------------------
                          (1 + 2*a)*(-1 + 2*a)
$$\frac{2 a \sin{\left(a x - \frac{x}{2} \right)} + 2 a \sin{\left(a x + \frac{x}{2} \right)} + \sin{\left(a x - \frac{x}{2} \right)} - \sin{\left(a x + \frac{x}{2} \right)}}{\left(2 a - 1\right) \left(2 a + 1\right)}$$ 
(-sin(x/2 + a*x) + 2*a*sin(x/2 + a*x) + 2*a*sin(-x/2 + a*x) + sin(-x/2 + a*x))/((1 + 2*a)*(-1 + 2*a))
Unión de expresiones racionales [src] 
             /x*(1 + 2*a)\                /x*(1 - 2*a)\
(1 - 2*a)*sin|-----------| + (1 + 2*a)*sin|-----------|
             \     2     /                \     2     /
-------------------------------------------------------
                  (1 - 2*a)*(1 + 2*a)
$$\frac{\left(1 - 2 a\right) \sin{\left(\frac{x \left(2 a + 1\right)}{2} \right)} + \left(2 a + 1\right) \sin{\left(\frac{x \left(1 - 2 a\right)}{2} \right)}}{\left(1 - 2 a\right) \left(2 a + 1\right)}$$ 
((1 - 2*a)*sin(x*(1 + 2*a)/2) + (1 + 2*a)*sin(x*(1 - 2*a)/2))/((1 - 2*a)*(1 + 2*a))
Compilar la expresión [src] 
sin((1/2 - a)*x)   sin((a + 1/2)*x)
---------------- + ----------------
    1 - 2*a            1 + 2*a
$$\frac{\sin{\left(x \left(a + \frac{1}{2}\right) \right)}}{2 a + 1} + \frac{\sin{\left(x \left(\frac{1}{2} - a\right) \right)}}{1 - 2 a}$$ 
sin((1/2 - a)*x)/(1 - 2*a) + sin((a + 1/2)*x)/(1 + 2*a)
Abrimos la expresión [src] 
            /x\      /x\                        /x\      /x\         
cos(a*x)*sin|-|   cos|-|*sin(a*x)   cos(a*x)*sin|-|   cos|-|*sin(a*x)
            \2/      \2/                        \2/      \2/         
--------------- + --------------- + --------------- - ---------------
    1 - 2*a           1 + 2*a           1 + 2*a           1 - 2*a
$$\frac{\sin{\left(\frac{x}{2} \right)} \cos{\left(a x \right)}}{2 a + 1} + \frac{\sin{\left(a x \right)} \cos{\left(\frac{x}{2} \right)}}{2 a + 1} + \frac{\sin{\left(\frac{x}{2} \right)} \cos{\left(a x \right)}}{1 - 2 a} - \frac{\sin{\left(a x \right)} \cos{\left(\frac{x}{2} \right)}}{1 - 2 a}$$ 
cos(a*x)*sin(x/2)/(1 - 2*a) + cos(x/2)*sin(a*x)/(1 + 2*a) + cos(a*x)*sin(x/2)/(1 + 2*a) - cos(x/2)*sin(a*x)/(1 - 2*a)
Parte trigonométrica [src] 
            1                             1             
-------------------------- - ---------------------------
(1 + 2*a)*csc(x*(1/2 + a))   (1 - 2*a)*csc(x*(-1/2 + a))
$$\frac{1}{\left(2 a + 1\right) \csc{\left(x \left(a + \frac{1}{2}\right) \right)}} - \frac{1}{\left(1 - 2 a\right) \csc{\left(x \left(a - \frac{1}{2}\right) \right)}}$$ 
   /  pi              \      /  pi              \
cos|- -- + x*(1/2 - a)|   cos|- -- + x*(1/2 + a)|
   \  2               /      \  2               /
----------------------- + -----------------------
        1 - 2*a                   1 + 2*a
$$\frac{\cos{\left(x \left(a + \frac{1}{2}\right) - \frac{\pi}{2} \right)}}{2 a + 1} + \frac{\cos{\left(x \left(\frac{1}{2} - a\right) - \frac{\pi}{2} \right)}}{1 - 2 a}$$ 
sin(x*(1/2 + a))   sin(x*(-1/2 + a))
---------------- - -----------------
    1 + 2*a             1 - 2*a
$$\frac{\sin{\left(x \left(a + \frac{1}{2}\right) \right)}}{2 a + 1} - \frac{\sin{\left(x \left(a - \frac{1}{2}\right) \right)}}{1 - 2 a}$$ 
            1                            1             
-------------------------- + --------------------------
(1 - 2*a)*csc(x*(1/2 - a))   (1 + 2*a)*csc(x*(1/2 + a))
$$\frac{1}{\left(2 a + 1\right) \csc{\left(x \left(a + \frac{1}{2}\right) \right)}} + \frac{1}{\left(1 - 2 a\right) \csc{\left(x \left(\frac{1}{2} - a\right) \right)}}$$ 
   /  pi              \      /  pi               \
cos|- -- + x*(1/2 + a)|   cos|- -- + x*(-1/2 + a)|
   \  2               /      \  2                /
----------------------- - ------------------------
        1 + 2*a                   1 - 2*a
$$\frac{\cos{\left(x \left(a + \frac{1}{2}\right) - \frac{\pi}{2} \right)}}{2 a + 1} - \frac{\cos{\left(x \left(a - \frac{1}{2}\right) - \frac{\pi}{2} \right)}}{1 - 2 a}$$ 
             /x*(1/2 + a)\                       /x*(1/2 - a)\       
        2*cot|-----------|                  2*cot|-----------|       
             \     2     /                       \     2     /       
--------------------------------- + ---------------------------------
/       2/x*(1/2 + a)\\             /       2/x*(1/2 - a)\\          
|1 + cot |-----------||*(1 + 2*a)   |1 + cot |-----------||*(1 - 2*a)
\        \     2     //             \        \     2     //
$$\frac{2 \cot{\left(\frac{x \left(a + \frac{1}{2}\right)}{2} \right)}}{\left(2 a + 1\right) \left(\cot^{2}{\left(\frac{x \left(a + \frac{1}{2}\right)}{2} \right)} + 1\right)} + \frac{2 \cot{\left(\frac{x \left(\frac{1}{2} - a\right)}{2} \right)}}{\left(1 - 2 a\right) \left(\cot^{2}{\left(\frac{x \left(\frac{1}{2} - a\right)}{2} \right)} + 1\right)}$$ 
              /x*(-1/2 + a)\                        /x*(1/2 + a)\       
         2*tan|------------|                   2*tan|-----------|       
              \     2      /                        \     2     /       
- ---------------------------------- + ---------------------------------
  /       2/x*(-1/2 + a)\\             /       2/x*(1/2 + a)\\          
  |1 + tan |------------||*(1 - 2*a)   |1 + tan |-----------||*(1 + 2*a)
  \        \     2      //             \        \     2     //
$$\frac{2 \tan{\left(\frac{x \left(a + \frac{1}{2}\right)}{2} \right)}}{\left(2 a + 1\right) \left(\tan^{2}{\left(\frac{x \left(a + \frac{1}{2}\right)}{2} \right)} + 1\right)} - \frac{2 \tan{\left(\frac{x \left(a - \frac{1}{2}\right)}{2} \right)}}{\left(1 - 2 a\right) \left(\tan^{2}{\left(\frac{x \left(a - \frac{1}{2}\right)}{2} \right)} + 1\right)}$$ 
                1                                   1                
--------------------------------- + ---------------------------------
             /  pi              \                /  pi              \
(1 - 2*a)*sec|- -- + x*(1/2 - a)|   (1 + 2*a)*sec|- -- + x*(1/2 + a)|
             \  2               /                \  2               /
$$\frac{1}{\left(2 a + 1\right) \sec{\left(x \left(a + \frac{1}{2}\right) - \frac{\pi}{2} \right)}} + \frac{1}{\left(1 - 2 a\right) \sec{\left(x \left(\frac{1}{2} - a\right) - \frac{\pi}{2} \right)}}$$ 
     1                              1                      
-----------*sin((1/2 - a)*x) + -----------*sin((a + 1/2)*x)
2*(1/2 - a)                    2*(a + 1/2)
$$\frac{1}{2 \left(\frac{1}{2} - a\right)} \sin{\left(x \left(\frac{1}{2} - a\right) \right)} + \frac{1}{2 \left(a + \frac{1}{2}\right)} \sin{\left(x \left(a + \frac{1}{2}\right) \right)}$$ 
              /x*(-1/2 + a)\                        /x*(1/2 + a)\       
         2*cot|------------|                   2*cot|-----------|       
              \     2      /                        \     2     /       
- ---------------------------------- + ---------------------------------
  /       2/x*(-1/2 + a)\\             /       2/x*(1/2 + a)\\          
  |1 + cot |------------||*(1 - 2*a)   |1 + cot |-----------||*(1 + 2*a)
  \        \     2      //             \        \     2     //
$$\frac{2 \cot{\left(\frac{x \left(a + \frac{1}{2}\right)}{2} \right)}}{\left(2 a + 1\right) \left(\cot^{2}{\left(\frac{x \left(a + \frac{1}{2}\right)}{2} \right)} + 1\right)} - \frac{2 \cot{\left(\frac{x \left(a - \frac{1}{2}\right)}{2} \right)}}{\left(1 - 2 a\right) \left(\cot^{2}{\left(\frac{x \left(a - \frac{1}{2}\right)}{2} \right)} + 1\right)}$$ 
sin(x*(1/2 - a))   sin(x*(1/2 + a))
---------------- + ----------------
    1 - 2*a            1 + 2*a
$$\frac{\sin{\left(x \left(a + \frac{1}{2}\right) \right)}}{2 a + 1} + \frac{\sin{\left(x \left(\frac{1}{2} - a\right) \right)}}{1 - 2 a}$$ 
                1                                   1                 
--------------------------------- - ----------------------------------
             /  pi              \                /  pi               \
(1 + 2*a)*sec|- -- + x*(1/2 + a)|   (1 - 2*a)*sec|- -- + x*(-1/2 + a)|
             \  2               /                \  2                /
$$\frac{1}{\left(2 a + 1\right) \sec{\left(x \left(a + \frac{1}{2}\right) - \frac{\pi}{2} \right)}} - \frac{1}{\left(1 - 2 a\right) \sec{\left(x \left(a - \frac{1}{2}\right) - \frac{\pi}{2} \right)}}$$ 
             /x*(1/2 + a)\                       /x*(1/2 - a)\       
        2*tan|-----------|                  2*tan|-----------|       
             \     2     /                       \     2     /       
--------------------------------- + ---------------------------------
/       2/x*(1/2 + a)\\             /       2/x*(1/2 - a)\\          
|1 + tan |-----------||*(1 + 2*a)   |1 + tan |-----------||*(1 - 2*a)
\        \     2     //             \        \     2     //
$$\frac{2 \tan{\left(\frac{x \left(a + \frac{1}{2}\right)}{2} \right)}}{\left(2 a + 1\right) \left(\tan^{2}{\left(\frac{x \left(a + \frac{1}{2}\right)}{2} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{x \left(\frac{1}{2} - a\right)}{2} \right)}}{\left(1 - 2 a\right) \left(\tan^{2}{\left(\frac{x \left(\frac{1}{2} - a\right)}{2} \right)} + 1\right)}$$ 
2*tan(x*(1/2 + a)/2)/((1 + tan(x*(1/2 + a)/2)^2)*(1 + 2*a)) + 2*tan(x*(1/2 - a)/2)/((1 + tan(x*(1/2 - a)/2)^2)*(1 - 2*a))
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