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¿Cómo vas a descomponer esta tan(x)/(2+2*tan(x)^2)+x/2 expresión en fracciones?

Expresión a simplificar:

Solución

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