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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(x)/((tan(pi/4)-(x/2))*(1+sin(x)))

¿Cómo vas a descomponer esta sin(x)/((tan(pi/4)-(x/2))*(1+sin(x))) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
          sin(x)          
--------------------------
/   /pi\   x\             
|tan|--| - -|*(1 + sin(x))
\   \4 /   2/
sin⁡(x)(−x2+tan⁡(π4))(sin⁡(x)+1)\frac{\sin{\left(x \right)}}{\left(- \frac{x}{2} + \tan{\left(\frac{\pi}{4} \right)}\right) \left(\sin{\left(x \right)} + 1\right)} 
sin(x)/(((tan(pi/4) - x/2)*(1 + sin(x))))
Simplificación general [src] 
      -2*sin(x)      
---------------------
(1 + sin(x))*(-2 + x)
−2sin⁡(x)(x−2)(sin⁡(x)+1)- \frac{2 \sin{\left(x \right)}}{\left(x - 2\right) \left(\sin{\left(x \right)} + 1\right)} 
-2*sin(x)/((1 + sin(x))*(-2 + x))
Denominador común [src] 
         -2*sin(x)          
----------------------------
-2 + x - 2*sin(x) + x*sin(x)
−2sin⁡(x)xsin⁡(x)+x−2sin⁡(x)−2- \frac{2 \sin{\left(x \right)}}{x \sin{\left(x \right)} + x - 2 \sin{\left(x \right)} - 2} 
-2*sin(x)/(-2 + x - 2*sin(x) + x*sin(x))
Denominador racional [src] 
           2*sin(x)          
-----------------------------
             /          /pi\\
(1 + sin(x))*|-x + 2*tan|--||
             \          \4 //
2sin⁡(x)(−x+2tan⁡(π4))(sin⁡(x)+1)\frac{2 \sin{\left(x \right)}}{\left(- x + 2 \tan{\left(\frac{\pi}{4} \right)}\right) \left(\sin{\left(x \right)} + 1\right)} 
2*sin(x)/((1 + sin(x))*(-x + 2*tan(pi/4)))
Combinatoria [src] 
      -2*sin(x)      
---------------------
(1 + sin(x))*(-2 + x)
−2sin⁡(x)(x−2)(sin⁡(x)+1)- \frac{2 \sin{\left(x \right)}}{\left(x - 2\right) \left(\sin{\left(x \right)} + 1\right)} 
-2*sin(x)/((1 + sin(x))*(-2 + x))
Respuesta numérica [src] 
sin(x)/((1.0 - 0.5*x)*(1.0 + sin(x)))
sin(x)/((1.0 - 0.5*x)*(1.0 + sin(x)))
Unión de expresiones racionales [src] 
      2*sin(x)      
--------------------
(1 + sin(x))*(2 - x)
2sin⁡(x)(2−x)(sin⁡(x)+1)\frac{2 \sin{\left(x \right)}}{\left(2 - x\right) \left(\sin{\left(x \right)} + 1\right)} 
2*sin(x)/((1 + sin(x))*(2 - x))
Potencias [src] 
                     /   -I*x    I*x\                   
                  -I*\- e     + e   /                   
--------------------------------------------------------
                           /        /   pi*I    -pi*I \\
                           |        |   ----    ------||
  /      /   -I*x    I*x\\ |        |    4        4   ||
  |    I*\- e     + e   /| |  x   I*\- e     + e      /|
2*|1 - ------------------|*|- - + ---------------------|
  \            2         / |  2       -pi*I     pi*I   |
                           |          ------    ----   |
                           |            4        4     |
                           \         e       + e       /
−i(eix−e−ix)2(−x2+i(−eiπ4+e−iπ4)e−iπ4+eiπ4)(−i(eix−e−ix)2+1)- \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(- \frac{x}{2} + \frac{i \left(- e^{\frac{i \pi}{4}} + e^{- \frac{i \pi}{4}}\right)}{e^{- \frac{i \pi}{4}} + e^{\frac{i \pi}{4}}}\right) \left(- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 1\right)} 
       sin(x)       
--------------------
/    x\             
|1 - -|*(1 + sin(x))
\    2/
sin⁡(x)(1−x2)(sin⁡(x)+1)\frac{\sin{\left(x \right)}}{\left(1 - \frac{x}{2}\right) \left(\sin{\left(x \right)} + 1\right)} 
sin(x)/((1 - x/2)*(1 + sin(x)))
Abrimos la expresión [src] 
          sin(x)         
-------------------------
    x   x*sin(x)         
1 - - - -------- + sin(x)
    2      2
sin⁡(x)−xsin⁡(x)2−x2+sin⁡(x)+1\frac{\sin{\left(x \right)}}{- \frac{x \sin{\left(x \right)}}{2} - \frac{x}{2} + \sin{\left(x \right)} + 1} 
sin(x)/(1 - x/2 - x*sin(x)/2 + sin(x))
Parte trigonométrica [src] 
                  1                  
-------------------------------------
/         1     \ /    x\    /    pi\
|1 + -----------|*|1 - -|*sec|x - --|
|       /    pi\| \    2/    \    2 /
|    sec|x - --||                    
\       \    2 //
1(1−x2)(1+1sec⁡(x−π2))sec⁡(x−π2)\frac{1}{\left(1 - \frac{x}{2}\right) \left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}} 
       sin(x)       
--------------------
/    x\             
|1 - -|*(1 + sin(x))
\    2/
sin⁡(x)(−x2+1)(sin⁡(x)+1)\frac{\sin{\left(x \right)}}{\left(- \frac{x}{2} + 1\right) \left(\sin{\left(x \right)} + 1\right)} 
          /    pi\       
       cos|x - --|       
          \    2 /       
-------------------------
/    x\ /       /    pi\\
|1 - -|*|1 + cos|x - --||
\    2/ \       \    2 //
cos⁡(x−π2)(1−x2)(cos⁡(x−π2)+1)\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\left(1 - \frac{x}{2}\right) \left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)} 
       sin(x)       
--------------------
/    x\             
|1 - -|*(1 + sin(x))
\    2/
sin⁡(x)(1−x2)(sin⁡(x)+1)\frac{\sin{\left(x \right)}}{\left(1 - \frac{x}{2}\right) \left(\sin{\left(x \right)} + 1\right)} 
                     /x\               
                2*tan|-|               
                     \2/               
---------------------------------------
                      /           /x\ \
                      |      2*tan|-| |
/       2/x\\ /    x\ |           \2/ |
|1 + tan |-||*|1 - -|*|1 + -----------|
\        \2// \    2/ |           2/x\|
                      |    1 + tan |-||
                      \            \2//
2tan⁡(x2)(1−x2)(1+2tan⁡(x2)tan⁡2(x2)+1)(tan⁡2(x2)+1)\frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(1 - \frac{x}{2}\right) \left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
                     /x\               
                2*cot|-|               
                     \2/               
---------------------------------------
                      /           /x\ \
                      |      2*cot|-| |
/       2/x\\ /    x\ |           \2/ |
|1 + cot |-||*|1 - -|*|1 + -----------|
\        \2// \    2/ |           2/x\|
                      |    1 + cot |-||
                      \            \2//
2cot⁡(x2)(1−x2)(1+2cot⁡(x2)cot⁡2(x2)+1)(cot⁡2(x2)+1)\frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(1 - \frac{x}{2}\right) \left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
                  /    pi\               
               cos|x - --|               
                  \    2 /               
-----------------------------------------
/       /    pi\\ /  x     ___    /-pi \\
|1 + cos|x - --||*|- - + \/ 2 *cos|----||
\       \    2 // \  2            \ 4  //
cos⁡(x−π2)(−x2+2cos⁡(−π4))(cos⁡(x−π2)+1)\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\left(- \frac{x}{2} + \sqrt{2} \cos{\left(- \frac{\pi}{4} \right)}\right) \left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)} 
             1             
---------------------------
/      1   \ /    x\       
|1 + ------|*|1 - -|*csc(x)
\    csc(x)/ \    2/
1(1−x2)(1+1csc⁡(x))csc⁡(x)\frac{1}{\left(1 - \frac{x}{2}\right) \left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}} 
1/((1 + 1/csc(x))*(1 - x/2)*csc(x))
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