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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(6*k*x+x+pi/6)/(2*(1+6*k))+sin(6*k*x-x-pi/6)/(2*(6*k-1))

¿Cómo vas a descomponer esta sin(6*k*x+x+pi/6)/(2*(1+6*k))+sin(6*k*x-x-pi/6)/(2*(6*k-1)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   /            pi\      /            pi\
sin|6*k*x + x + --|   sin|6*k*x - x - --|
   \            6 /      \            6 /
------------------- + -------------------
    2*(1 + 6*k)           2*(6*k - 1)
$$\frac{\sin{\left(\left(6 k x + x\right) + \frac{\pi}{6} \right)}}{2 \left(6 k + 1\right)} + \frac{\sin{\left(\left(6 k x - x\right) - \frac{\pi}{6} \right)}}{2 \left(6 k - 1\right)}$$ 
sin((6*k)*x + x + pi/6)/((2*(1 + 6*k))) + sin((6*k)*x - x - pi/6)/((2*(6*k - 1)))
Simplificación general [src] 
              /    pi        \                /    pi        \
(-1 + 6*k)*sin|x + -- + 6*k*x| - (1 + 6*k)*sin|x + -- - 6*k*x|
              \    6         /                \    6         /
--------------------------------------------------------------
                    2*(1 + 6*k)*(-1 + 6*k)
$$\frac{\left(6 k - 1\right) \sin{\left(6 k x + x + \frac{\pi}{6} \right)} - \left(6 k + 1\right) \sin{\left(- 6 k x + x + \frac{\pi}{6} \right)}}{2 \left(6 k - 1\right) \left(6 k + 1\right)}$$ 
((-1 + 6*k)*sin(x + pi/6 + 6*k*x) - (1 + 6*k)*sin(x + pi/6 - 6*k*x))/(2*(1 + 6*k)*(-1 + 6*k))
Respuesta numérica [src] 
sin((6*k)*x + x + pi/6)/(2.0 + 12.0*k) + sin((6*k)*x - x - pi/6)/(-2.0 + 12.0*k)
sin((6*k)*x + x + pi/6)/(2.0 + 12.0*k) + sin((6*k)*x - x - pi/6)/(-2.0 + 12.0*k)
Denominador común [src] 
 /         /    pi        \          /    pi        \      /    pi        \      /    pi        \\ 
-|- 6*k*sin|x + -- + 6*k*x| + 6*k*sin|x + -- - 6*k*x| + sin|x + -- - 6*k*x| + sin|x + -- + 6*k*x|| 
 \         \    6         /          \    6         /      \    6         /      \    6         // 
---------------------------------------------------------------------------------------------------
                                                      2                                            
                                             -2 + 72*k
$$- \frac{6 k \sin{\left(- 6 k x + x + \frac{\pi}{6} \right)} - 6 k \sin{\left(6 k x + x + \frac{\pi}{6} \right)} + \sin{\left(- 6 k x + x + \frac{\pi}{6} \right)} + \sin{\left(6 k x + x + \frac{\pi}{6} \right)}}{72 k^{2} - 2}$$ 
-(-6*k*sin(x + pi/6 + 6*k*x) + 6*k*sin(x + pi/6 - 6*k*x) + sin(x + pi/6 - 6*k*x) + sin(x + pi/6 + 6*k*x))/(-2 + 72*k^2)
Potencias [src] 
   /    pi        \      /    pi        \
sin|x + -- + 6*k*x|   sin|x + -- - 6*k*x|
   \    6         /      \    6         /
------------------- - -------------------
      2 + 12*k             -2 + 12*k
$$\frac{\sin{\left(6 k x + x + \frac{\pi}{6} \right)}}{12 k + 2} - \frac{\sin{\left(- 6 k x + x + \frac{\pi}{6} \right)}}{12 k - 2}$$ 
    /     /    pi        \      /     pi        \\     /     /     pi        \      /    pi        \\
    |   I*|x + -- - 6*k*x|    I*|-x - -- + 6*k*x||     |   I*|-x - -- - 6*k*x|    I*|x + -- + 6*k*x||
    |     \    6         /      \     6         /|     |     \     6         /      \    6         /|
  I*\- e                   + e                   /   I*\- e                    + e                  /
- ------------------------------------------------ - ------------------------------------------------
                   2*(-2 + 12*k)                                       2*(2 + 12*k)
$$- \frac{i \left(- e^{i \left(- 6 k x - x - \frac{\pi}{6}\right)} + e^{i \left(6 k x + x + \frac{\pi}{6}\right)}\right)}{2 \left(12 k + 2\right)} - \frac{i \left(- e^{i \left(- 6 k x + x + \frac{\pi}{6}\right)} + e^{i \left(6 k x - x - \frac{\pi}{6}\right)}\right)}{2 \left(12 k - 2\right)}$$ 
-i*(-exp(i*(x + pi/6 - 6*k*x)) + exp(i*(-x - pi/6 + 6*k*x)))/(2*(-2 + 12*k)) - i*(-exp(i*(-x - pi/6 - 6*k*x)) + exp(i*(x + pi/6 + 6*k*x)))/(2*(2 + 12*k))
Denominador racional [src] 
       /    pi        \        /    pi        \           /    pi        \           /    pi        \
- 2*sin|x + -- - 6*k*x| - 2*sin|x + -- + 6*k*x| - 12*k*sin|x + -- - 6*k*x| + 12*k*sin|x + -- + 6*k*x|
       \    6         /        \    6         /           \    6         /           \    6         /
-----------------------------------------------------------------------------------------------------
                                        (-2 + 12*k)*(2 + 12*k)
$$\frac{- 12 k \sin{\left(- 6 k x + x + \frac{\pi}{6} \right)} + 12 k \sin{\left(6 k x + x + \frac{\pi}{6} \right)} - 2 \sin{\left(- 6 k x + x + \frac{\pi}{6} \right)} - 2 \sin{\left(6 k x + x + \frac{\pi}{6} \right)}}{\left(12 k - 2\right) \left(12 k + 2\right)}$$ 
(-2*sin(x + pi/6 - 6*k*x) - 2*sin(x + pi/6 + 6*k*x) - 12*k*sin(x + pi/6 - 6*k*x) + 12*k*sin(x + pi/6 + 6*k*x))/((-2 + 12*k)*(2 + 12*k))
Unión de expresiones racionales [src] 
             /-pi + 6*x*(-1 + 6*k)\                 /pi + 6*x*(1 + 6*k)\
(1 + 6*k)*sin|--------------------| + (-1 + 6*k)*sin|------------------|
             \         6          /                 \        6         /
------------------------------------------------------------------------
                         2*(1 + 6*k)*(-1 + 6*k)
$$\frac{\left(6 k - 1\right) \sin{\left(\frac{6 x \left(6 k + 1\right) + \pi}{6} \right)} + \left(6 k + 1\right) \sin{\left(\frac{6 x \left(6 k - 1\right) - \pi}{6} \right)}}{2 \left(6 k - 1\right) \left(6 k + 1\right)}$$ 
((1 + 6*k)*sin((-pi + 6*x*(-1 + 6*k))/6) + (-1 + 6*k)*sin((pi + 6*x*(1 + 6*k))/6))/(2*(1 + 6*k)*(-1 + 6*k))
Combinatoria [src] 
 /         /    pi        \          /    pi        \      /    pi        \      /    pi        \\ 
-|- 6*k*sin|x + -- + 6*k*x| + 6*k*sin|x + -- - 6*k*x| + sin|x + -- - 6*k*x| + sin|x + -- + 6*k*x|| 
 \         \    6         /          \    6         /      \    6         /      \    6         // 
---------------------------------------------------------------------------------------------------
                                       2*(1 + 6*k)*(-1 + 6*k)
$$- \frac{6 k \sin{\left(- 6 k x + x + \frac{\pi}{6} \right)} - 6 k \sin{\left(6 k x + x + \frac{\pi}{6} \right)} + \sin{\left(- 6 k x + x + \frac{\pi}{6} \right)} + \sin{\left(6 k x + x + \frac{\pi}{6} \right)}}{2 \left(6 k - 1\right) \left(6 k + 1\right)}$$ 
-(-6*k*sin(x + pi/6 + 6*k*x) + 6*k*sin(x + pi/6 - 6*k*x) + sin(x + pi/6 - 6*k*x) + sin(x + pi/6 + 6*k*x))/(2*(1 + 6*k)*(-1 + 6*k))
Abrimos la expresión [src] 
                                                                                  ___                       ___                       ___                       ___                  
cos(x)*cos(6*k*x)   cos(x)*cos(6*k*x)   sin(x)*sin(6*k*x)   sin(x)*sin(6*k*x)   \/ 3 *cos(x)*sin(6*k*x)   \/ 3 *cos(x)*sin(6*k*x)   \/ 3 *cos(6*k*x)*sin(x)   \/ 3 *cos(6*k*x)*sin(x)
----------------- - ----------------- - ----------------- - ----------------- + ----------------------- + ----------------------- + ----------------------- - -----------------------
   2*(2 + 12*k)       2*(-2 + 12*k)       2*(-2 + 12*k)        2*(2 + 12*k)          2*(-2 + 12*k)              2*(2 + 12*k)              2*(2 + 12*k)             2*(-2 + 12*k)
$$- \frac{\sin{\left(x \right)} \sin{\left(6 k x \right)}}{2 \left(12 k + 2\right)} + \frac{\sqrt{3} \sin{\left(x \right)} \cos{\left(6 k x \right)}}{2 \left(12 k + 2\right)} + \frac{\sqrt{3} \sin{\left(6 k x \right)} \cos{\left(x \right)}}{2 \left(12 k + 2\right)} + \frac{\cos{\left(x \right)} \cos{\left(6 k x \right)}}{2 \left(12 k + 2\right)} - \frac{\sin{\left(x \right)} \sin{\left(6 k x \right)}}{2 \left(12 k - 2\right)} - \frac{\sqrt{3} \sin{\left(x \right)} \cos{\left(6 k x \right)}}{2 \left(12 k - 2\right)} + \frac{\sqrt{3} \sin{\left(6 k x \right)} \cos{\left(x \right)}}{2 \left(12 k - 2\right)} - \frac{\cos{\left(x \right)} \cos{\left(6 k x \right)}}{2 \left(12 k - 2\right)}$$ 
cos(x)*cos((6*k)*x)/(2*(2 + 12*k)) - cos(x)*cos((6*k)*x)/(2*(-2 + 12*k)) - sin(x)*sin((6*k)*x)/(2*(-2 + 12*k)) - sin(x)*sin((6*k)*x)/(2*(2 + 12*k)) + sqrt(3)*cos(x)*sin((6*k)*x)/(2*(-2 + 12*k)) + sqrt(3)*cos(x)*sin((6*k)*x)/(2*(2 + 12*k)) + sqrt(3)*cos((6*k)*x)*sin(x)/(2*(2 + 12*k)) - sqrt(3)*cos((6*k)*x)*sin(x)/(2*(-2 + 12*k))
Compilar la expresión [src] 
   /            pi\      /            pi\
sin|6*k*x - x - --|   sin|6*k*x + x + --|
   \            6 /      \            6 /
------------------- + -------------------
     -2 + 12*k              2 + 12*k
$$\frac{\sin{\left(\left(6 k x + x\right) + \frac{\pi}{6} \right)}}{12 k + 2} + \frac{\sin{\left(\left(6 k x - x\right) - \frac{\pi}{6} \right)}}{12 k - 2}$$ 
sin((6*k)*x - x - pi/6)/(-2 + 12*k) + sin((6*k)*x + x + pi/6)/(2 + 12*k)
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