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Simplificación de expresiones/ Descomposición en fracciones simples/ (sin(4*x)/((2+2*x)*8)+sin(2*x)/2+x/2)/2

¿Cómo vas a descomponer esta (sin(4*x)/((2+2*x)*8)+sin(2*x)/2+x/2)/2 expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
  sin(4*x)    sin(2*x)   x
----------- + -------- + -
(2 + 2*x)*8      2       2
--------------------------
            2
$$\frac{\frac{x}{2} + \left(\frac{\sin{\left(2 x \right)}}{2} + \frac{\sin{\left(4 x \right)}}{8 \left(2 x + 2\right)}\right)}{2}$$ 
(sin(4*x)/(((2 + 2*x)*8)) + sin(2*x)/2 + x/2)/2
Simplificación general [src] 
(8 + 8*x)*sin(2*x) + 8*x*(1 + x) + sin(4*x)
-------------------------------------------
                 32*(1 + x)
$$\frac{8 x \left(x + 1\right) + \left(8 x + 8\right) \sin{\left(2 x \right)} + \sin{\left(4 x \right)}}{32 \left(x + 1\right)}$$ 
((8 + 8*x)*sin(2*x) + 8*x*(1 + x) + sin(4*x))/(32*(1 + x))
Respuesta numérica [src] 
0.25*x + 0.25*sin(2*x) + 0.5*sin(4*x)/(16.0 + 16.0*x)
0.25*x + 0.25*sin(2*x) + 0.5*sin(4*x)/(16.0 + 16.0*x)
Denominador racional [src] 
4*sin(4*x) + x*(32 + 32*x) + 2*(16 + 16*x)*sin(2*x)
---------------------------------------------------
                    128 + 128*x
$$\frac{x \left(32 x + 32\right) + 2 \left(16 x + 16\right) \sin{\left(2 x \right)} + 4 \sin{\left(4 x \right)}}{128 x + 128}$$ 
(4*sin(4*x) + x*(32 + 32*x) + 2*(16 + 16*x)*sin(2*x))/(128 + 128*x)
Abrimos la expresión [src] 
x   sin(2*x)     sin(4*x)  
- + -------- + ------------
4      4       16*(2 + 2*x)
$$\frac{x}{4} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{16 \left(2 x + 2\right)}$$ 
                         3                            
x   cos(x)*sin(x)   4*sin (x)*cos(x)   2*cos(x)*sin(x)
- + ------------- - ---------------- + ---------------
4         2            16 + 16*x          16 + 16*x
$$\frac{x}{4} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2} - \frac{4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{16 x + 16} + \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{16 x + 16}$$ 
x/4 + cos(x)*sin(x)/2 - 4*sin(x)^3*cos(x)/(16 + 16*x) + 2*cos(x)*sin(x)/(16 + 16*x)
Denominador común [src] 
x   sin(2*x)    sin(4*x)
- + -------- + ---------
4      4       32 + 32*x
$$\frac{x}{4} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32 x + 32}$$ 
x/4 + sin(2*x)/4 + sin(4*x)/(32 + 32*x)
Compilar la expresión [src] 
x   sin(2*x)      sin(4*x)  
- + -------- + -------------
4      4       2*(16 + 16*x)
$$\frac{x}{4} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{2 \left(16 x + 16\right)}$$ 
x/4 + sin(2*x)/4 + sin(4*x)/(2*(16 + 16*x))
Potencias [src] 
      /   -2*I*x    2*I*x\     /   -4*I*x    4*I*x\
x   I*\- e       + e     /   I*\- e       + e     /
- - ---------------------- - ----------------------
4             8                  4*(16 + 16*x)
$$\frac{x}{4} - \frac{i \left(e^{2 i x} - e^{- 2 i x}\right)}{8} - \frac{i \left(e^{4 i x} - e^{- 4 i x}\right)}{4 \left(16 x + 16\right)}$$ 
x   sin(2*x)      sin(4*x)  
- + -------- + -------------
4      4       2*(16 + 16*x)
$$\frac{x}{4} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{2 \left(16 x + 16\right)}$$ 
x/4 + sin(2*x)/4 + sin(4*x)/(2*(16 + 16*x))
Parte trigonométrica [src] 
       /      pi\      /      pi\
    cos|2*x - --|   cos|4*x - --|
x      \      2 /      \      2 /
- + ------------- + -------------
4         4         2*(16 + 16*x)
$$\frac{x}{4} + \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{4} + \frac{\cos{\left(4 x - \frac{\pi}{2} \right)}}{2 \left(16 x + 16\right)}$$ 
x        cot(x)                 cot(2*x)         
- + --------------- + ---------------------------
4     /       2   \   /       2     \            
    2*\1 + cot (x)/   \1 + cot (2*x)/*(16 + 16*x)
$$\frac{x}{4} + \frac{\cot{\left(x \right)}}{2 \left(\cot^{2}{\left(x \right)} + 1\right)} + \frac{\cot{\left(2 x \right)}}{\left(16 x + 16\right) \left(\cot^{2}{\left(2 x \right)} + 1\right)}$$ 
x          1                       1             
- + --------------- + ---------------------------
4        /      pi\                    /      pi\
    4*sec|2*x - --|   2*(16 + 16*x)*sec|4*x - --|
         \      2 /                    \      2 /
$$\frac{x}{4} + \frac{1}{4 \sec{\left(2 x - \frac{\pi}{2} \right)}} + \frac{1}{2 \left(16 x + 16\right) \sec{\left(4 x - \frac{\pi}{2} \right)}}$$ 
x   sin(2*x)     sin(4*x)  
- + -------- + ------------
4      4       16*(2 + 2*x)
$$\frac{x}{4} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{16 \left(2 x + 2\right)}$$ 
x   sin(2*x)      sin(4*x)  
- + -------- + -------------
4      4       2*(16 + 16*x)
$$\frac{x}{4} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{2 \left(16 x + 16\right)}$$ 
x        tan(x)                 tan(2*x)         
- + --------------- + ---------------------------
4     /       2   \   /       2     \            
    2*\1 + tan (x)/   \1 + tan (2*x)/*(16 + 16*x)
$$\frac{x}{4} + \frac{\tan{\left(x \right)}}{2 \left(\tan^{2}{\left(x \right)} + 1\right)} + \frac{\tan{\left(2 x \right)}}{\left(16 x + 16\right) \left(\tan^{2}{\left(2 x \right)} + 1\right)}$$ 
x       1                  1           
- + ---------- + ----------------------
4   4*csc(2*x)   2*(16 + 16*x)*csc(4*x)
$$\frac{x}{4} + \frac{1}{4 \csc{\left(2 x \right)}} + \frac{1}{2 \left(16 x + 16\right) \csc{\left(4 x \right)}}$$ 
x/4 + 1/(4*csc(2*x)) + 1/(2*(16 + 16*x)*csc(4*x))
Unión de expresiones racionales [src] 
8*x*(1 + x) + 8*(1 + x)*sin(2*x) + sin(4*x)
-------------------------------------------
                 32*(1 + x)
$$\frac{8 x \left(x + 1\right) + 8 \left(x + 1\right) \sin{\left(2 x \right)} + \sin{\left(4 x \right)}}{32 \left(x + 1\right)}$$ 
(8*x*(1 + x) + 8*(1 + x)*sin(2*x) + sin(4*x))/(32*(1 + x))
Combinatoria [src] 
         2                                       
8*x + 8*x  + 8*sin(2*x) + 8*x*sin(2*x) + sin(4*x)
-------------------------------------------------
                    32*(1 + x)
$$\frac{8 x^{2} + 8 x \sin{\left(2 x \right)} + 8 x + 8 \sin{\left(2 x \right)} + \sin{\left(4 x \right)}}{32 \left(x + 1\right)}$$ 
(8*x + 8*x^2 + 8*sin(2*x) + 8*x*sin(2*x) + sin(4*x))/(32*(1 + x))
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