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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(k*x-5*x)/(k-5)-sin(k*x+6*x)/(6+k)

¿Cómo vas a descomponer esta sin(k*x-5*x)/(k-5)-sin(k*x+6*x)/(6+k) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
sin(k*x - 5*x)   sin(k*x + 6*x)
-------------- - --------------
    k - 5            6 + k
−sin⁡(kx+6x)k+6+sin⁡(kx−5x)k−5- \frac{\sin{\left(k x + 6 x \right)}}{k + 6} + \frac{\sin{\left(k x - 5 x \right)}}{k - 5} 
sin(k*x - 5*x)/(k - 5) - sin(k*x + 6*x)/(6 + k)
Simplificación general [src] 
(5 - k)*sin(x*(6 + k)) + (6 + k)*sin(x*(-5 + k))
------------------------------------------------
                (-5 + k)*(6 + k)
(5−k)sin⁡(x(k+6))+(k+6)sin⁡(x(k−5))(k−5)(k+6)\frac{\left(5 - k\right) \sin{\left(x \left(k + 6\right) \right)} + \left(k + 6\right) \sin{\left(x \left(k - 5\right) \right)}}{\left(k - 5\right) \left(k + 6\right)} 
((5 - k)*sin(x*(6 + k)) + (6 + k)*sin(x*(-5 + k)))/((-5 + k)*(6 + k))
Respuesta numérica [src] 
sin(k*x - 5*x)/(-5.0 + k) - sin(k*x + 6*x)/(6.0 + k)
sin(k*x - 5*x)/(-5.0 + k) - sin(k*x + 6*x)/(6.0 + k)
Denominador común [src] 
-(-6*sin(-5*x + k*x) - 5*sin(6*x + k*x) + k*sin(6*x + k*x) - k*sin(-5*x + k*x)) 
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                                             2                                  
                                  -30 + k + k
−−ksin⁡(kx−5x)+ksin⁡(kx+6x)−6sin⁡(kx−5x)−5sin⁡(kx+6x)k2+k−30- \frac{- k \sin{\left(k x - 5 x \right)} + k \sin{\left(k x + 6 x \right)} - 6 \sin{\left(k x - 5 x \right)} - 5 \sin{\left(k x + 6 x \right)}}{k^{2} + k - 30} 
-(-6*sin(-5*x + k*x) - 5*sin(6*x + k*x) + k*sin(6*x + k*x) - k*sin(-5*x + k*x))/(-30 + k + k^2)
Combinatoria [src] 
-(-6*sin(-5*x + k*x) - 5*sin(6*x + k*x) + k*sin(6*x + k*x) - k*sin(-5*x + k*x)) 
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                                (-5 + k)*(6 + k)
−−ksin⁡(kx−5x)+ksin⁡(kx+6x)−6sin⁡(kx−5x)−5sin⁡(kx+6x)(k−5)(k+6)- \frac{- k \sin{\left(k x - 5 x \right)} + k \sin{\left(k x + 6 x \right)} - 6 \sin{\left(k x - 5 x \right)} - 5 \sin{\left(k x + 6 x \right)}}{\left(k - 5\right) \left(k + 6\right)} 
-(-6*sin(-5*x + k*x) - 5*sin(6*x + k*x) + k*sin(6*x + k*x) - k*sin(-5*x + k*x))/((-5 + k)*(6 + k))
Denominador racional [src] 
5*sin(6*x + k*x) + 6*sin(-5*x + k*x) + k*sin(-5*x + k*x) - k*sin(6*x + k*x)
---------------------------------------------------------------------------
                              (-5 + k)*(6 + k)
ksin⁡(kx−5x)−ksin⁡(kx+6x)+6sin⁡(kx−5x)+5sin⁡(kx+6x)(k−5)(k+6)\frac{k \sin{\left(k x - 5 x \right)} - k \sin{\left(k x + 6 x \right)} + 6 \sin{\left(k x - 5 x \right)} + 5 \sin{\left(k x + 6 x \right)}}{\left(k - 5\right) \left(k + 6\right)} 
(5*sin(6*x + k*x) + 6*sin(-5*x + k*x) + k*sin(-5*x + k*x) - k*sin(6*x + k*x))/((-5 + k)*(6 + k))
Abrimos la expresión [src] 
                 6                     3                     2                     5                                                             5                     3                     4                     5                                                       3                   
sin(k*x)   32*cos (x)*sin(k*x)   20*cos (x)*sin(k*x)   18*cos (x)*sin(k*x)   16*sin (x)*cos(k*x)   5*cos(k*x)*sin(x)   5*cos(x)*sin(k*x)   16*cos (x)*sin(k*x)   20*sin (x)*cos(k*x)   48*cos (x)*sin(k*x)   32*sin (x)*cos(x)*cos(k*x)   6*cos(x)*cos(k*x)*sin(x)   32*sin (x)*cos(x)*cos(k*x)
-------- - ------------------- - ------------------- - ------------------- - ------------------- - ----------------- + ----------------- + ------------------- + ------------------- + ------------------- - -------------------------- - ------------------------ + --------------------------
 6 + k            6 + k                 k - 5                 6 + k                 k - 5                k - 5               k - 5                k - 5                 k - 5                 6 + k                    6 + k                       6 + k                       6 + k
−32sin⁡5(x)cos⁡(x)cos⁡(kx)k+6+32sin⁡3(x)cos⁡(x)cos⁡(kx)k+6−6sin⁡(x)cos⁡(x)cos⁡(kx)k+6−32sin⁡(kx)cos⁡6(x)k+6+48sin⁡(kx)cos⁡4(x)k+6−18sin⁡(kx)cos⁡2(x)k+6+sin⁡(kx)k+6−16sin⁡5(x)cos⁡(kx)k−5+20sin⁡3(x)cos⁡(kx)k−5−5sin⁡(x)cos⁡(kx)k−5+16sin⁡(kx)cos⁡5(x)k−5−20sin⁡(kx)cos⁡3(x)k−5+5sin⁡(kx)cos⁡(x)k−5- \frac{32 \sin^{5}{\left(x \right)} \cos{\left(x \right)} \cos{\left(k x \right)}}{k + 6} + \frac{32 \sin^{3}{\left(x \right)} \cos{\left(x \right)} \cos{\left(k x \right)}}{k + 6} - \frac{6 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(k x \right)}}{k + 6} - \frac{32 \sin{\left(k x \right)} \cos^{6}{\left(x \right)}}{k + 6} + \frac{48 \sin{\left(k x \right)} \cos^{4}{\left(x \right)}}{k + 6} - \frac{18 \sin{\left(k x \right)} \cos^{2}{\left(x \right)}}{k + 6} + \frac{\sin{\left(k x \right)}}{k + 6} - \frac{16 \sin^{5}{\left(x \right)} \cos{\left(k x \right)}}{k - 5} + \frac{20 \sin^{3}{\left(x \right)} \cos{\left(k x \right)}}{k - 5} - \frac{5 \sin{\left(x \right)} \cos{\left(k x \right)}}{k - 5} + \frac{16 \sin{\left(k x \right)} \cos^{5}{\left(x \right)}}{k - 5} - \frac{20 \sin{\left(k x \right)} \cos^{3}{\left(x \right)}}{k - 5} + \frac{5 \sin{\left(k x \right)} \cos{\left(x \right)}}{k - 5} 
sin(k*x)/(6 + k) - 32*cos(x)^6*sin(k*x)/(6 + k) - 20*cos(x)^3*sin(k*x)/(k - 5) - 18*cos(x)^2*sin(k*x)/(6 + k) - 16*sin(x)^5*cos(k*x)/(k - 5) - 5*cos(k*x)*sin(x)/(k - 5) + 5*cos(x)*sin(k*x)/(k - 5) + 16*cos(x)^5*sin(k*x)/(k - 5) + 20*sin(x)^3*cos(k*x)/(k - 5) + 48*cos(x)^4*sin(k*x)/(6 + k) - 32*sin(x)^5*cos(x)*cos(k*x)/(6 + k) - 6*cos(x)*cos(k*x)*sin(x)/(6 + k) + 32*sin(x)^3*cos(x)*cos(k*x)/(6 + k)
Unión de expresiones racionales [src] 
(6 + k)*sin(x*(-5 + k)) - (-5 + k)*sin(x*(6 + k))
-------------------------------------------------
                 (-5 + k)*(6 + k)
−(k−5)sin⁡(x(k+6))+(k+6)sin⁡(x(k−5))(k−5)(k+6)\frac{- \left(k - 5\right) \sin{\left(x \left(k + 6\right) \right)} + \left(k + 6\right) \sin{\left(x \left(k - 5\right) \right)}}{\left(k - 5\right) \left(k + 6\right)} 
((6 + k)*sin(x*(-5 + k)) - (-5 + k)*sin(x*(6 + k)))/((-5 + k)*(6 + k))
Potencias [src] 
  /   I*(-6*x - k*x)    I*(6*x + k*x)\     /   I*(5*x - k*x)    I*(-5*x + k*x)\
I*\- e               + e             /   I*\- e              + e              /
-------------------------------------- - --------------------------------------
              2*(6 + k)                                2*(-5 + k)
i(−ei(−kx−6x)+ei(kx+6x))2(k+6)−i(−ei(−kx+5x)+ei(kx−5x))2(k−5)\frac{i \left(- e^{i \left(- k x - 6 x\right)} + e^{i \left(k x + 6 x\right)}\right)}{2 \left(k + 6\right)} - \frac{i \left(- e^{i \left(- k x + 5 x\right)} + e^{i \left(k x - 5 x\right)}\right)}{2 \left(k - 5\right)} 
i*(-exp(i*(-6*x - k*x)) + exp(i*(6*x + k*x)))/(2*(6 + k)) - i*(-exp(i*(5*x - k*x)) + exp(i*(-5*x + k*x)))/(2*(-5 + k))
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