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Simplificación de expresiones/ Descomposición en fracciones simples/ sinx/(4-cos^2x)

¿Cómo vas a descomponer esta sinx/(4-cos^2x) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   sin(x)  
-----------
       2   
4 - cos (x)
sin⁡(x)4−cos⁡2(x)\frac{\sin{\left(x \right)}}{4 - \cos^{2}{\left(x \right)}} 
sin(x)/(4 - cos(x)^2)
Simplificación general [src] 
  -sin(x)   
------------
        2   
-4 + cos (x)
−sin⁡(x)cos⁡2(x)−4- \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)} - 4} 
-sin(x)/(-4 + cos(x)^2)
Denominador común [src] 
  -sin(x)   
------------
        2   
-4 + cos (x)
−sin⁡(x)cos⁡2(x)−4- \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)} - 4} 
-sin(x)/(-4 + cos(x)^2)
Combinatoria [src] 
         -sin(x)          
--------------------------
(-2 + cos(x))*(2 + cos(x))
−sin⁡(x)(cos⁡(x)−2)(cos⁡(x)+2)- \frac{\sin{\left(x \right)}}{\left(\cos{\left(x \right)} - 2\right) \left(\cos{\left(x \right)} + 2\right)} 
-sin(x)/((-2 + cos(x))*(2 + cos(x)))
Parte trigonométrica [src] 
            2             
--------------------------
/          1      \       
|7 - -------------|*csc(x)
|       /pi      \|       
|    csc|-- - 2*x||       
\       \2       //
2(7−1csc⁡(−2x+π2))csc⁡(x)\frac{2}{\left(7 - \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}\right) \csc{\left(x \right)}} 
   sin(x)  
-----------
       2   
3 + sin (x)
sin⁡(x)sin⁡2(x)+3\frac{\sin{\left(x \right)}}{\sin^{2}{\left(x \right)} + 3} 
     sin(x)     
----------------
       2/    pi\
4 - sin |x + --|
        \    2 /
sin⁡(x)4−sin⁡2(x+π2)\frac{\sin{\left(x \right)}}{4 - \sin^{2}{\left(x + \frac{\pi}{2} \right)}} 
                  /x\             
             2*cot|-|             
                  \2/             
----------------------------------
              /           2/x\   \
              |      4*cot |-|   |
/       2/x\\ |            \2/   |
|1 + cot |-||*|3 + --------------|
\        \2// |                 2|
              |    /       2/x\\ |
              |    |1 + cot |-|| |
              \    \        \2// /
2cot⁡(x2)(3+4cot⁡2(x2)(cot⁡2(x2)+1)2)(cot⁡2(x2)+1)\frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(3 + \frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
     /    pi\   
  cos|x - --|   
     \    2 /   
----------------
       2/    pi\
3 + cos |x - --|
        \    2 /
cos⁡(x−π2)cos⁡2(x−π2)+3\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)} + 3} 
   /    pi\
cos|x - --|
   \    2 /
-----------
       2   
4 - cos (x)
cos⁡(x−π2)4−cos⁡2(x)\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{4 - \cos^{2}{\left(x \right)}} 
         1          
--------------------
/       1   \       
|4 - -------|*csc(x)
|       2   |       
\    sec (x)/
1(4−1sec⁡2(x))csc⁡(x)\frac{1}{\left(4 - \frac{1}{\sec^{2}{\left(x \right)}}\right) \csc{\left(x \right)}} 
                 /x\           
            4*tan|-|           
                 \2/           
-------------------------------
              /           2   \
/       2/x\\ |    1 - tan (x)|
|1 + tan |-||*|7 - -----------|
\        \2// |           2   |
              \    1 + tan (x)/
4tan⁡(x2)(−1−tan⁡2(x)tan⁡2(x)+1+7)(tan⁡2(x2)+1)\frac{4 \tan{\left(\frac{x}{2} \right)}}{\left(- \frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} + 7\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
            2             
--------------------------
/       1    \    /    pi\
|7 - --------|*sec|x - --|
\    sec(2*x)/    \    2 /
2(7−1sec⁡(2x))sec⁡(x−π2)\frac{2}{\left(7 - \frac{1}{\sec{\left(2 x \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}} 
  2*sin(x)  
------------
7 - cos(2*x)
2sin⁡(x)7−cos⁡(2x)\frac{2 \sin{\left(x \right)}}{7 - \cos{\left(2 x \right)}} 
              1               
------------------------------
/         1      \    /    pi\
|3 + ------------|*sec|x - --|
|       2/    pi\|    \    2 /
|    sec |x - --||            
\        \    2 //
1(3+1sec⁡2(x−π2))sec⁡(x−π2)\frac{1}{\left(3 + \frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}} 
                   /x\             
              2*cot|-|             
                   \2/             
-----------------------------------
              /                  2\
              |    /        2/x\\ |
              |    |-1 + cot |-|| |
/       2/x\\ |    \         \2// |
|1 + cot |-||*|4 - ---------------|
\        \2// |                  2|
              |     /       2/x\\ |
              |     |1 + cot |-|| |
              \     \        \2// /
2cot⁡(x2)(−(cot⁡2(x2)−1)2(cot⁡2(x2)+1)2+4)(cot⁡2(x2)+1)\frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(- \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 4\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
     2*sin(x)    
-----------------
       /pi      \
7 - sin|-- + 2*x|
       \2       /
2sin⁡(x)7−sin⁡(2x+π2)\frac{2 \sin{\left(x \right)}}{7 - \sin{\left(2 x + \frac{\pi}{2} \right)}} 
            1            
-------------------------
/       1   \    /    pi\
|4 - -------|*sec|x - --|
|       2   |    \    2 /
\    sec (x)/
1(4−1sec⁡2(x))sec⁡(x−π2)\frac{1}{\left(4 - \frac{1}{\sec^{2}{\left(x \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}} 
         1          
--------------------
/       1   \       
|3 + -------|*csc(x)
|       2   |       
\    csc (x)/
1(3+1csc⁡2(x))csc⁡(x)\frac{1}{\left(3 + \frac{1}{\csc^{2}{\left(x \right)}}\right) \csc{\left(x \right)}} 
                  /x\             
             2*tan|-|             
                  \2/             
----------------------------------
              /                 2\
              |    /       2/x\\ |
              |    |1 - tan |-|| |
/       2/x\\ |    \        \2// |
|1 + tan |-||*|4 - --------------|
\        \2// |                 2|
              |    /       2/x\\ |
              |    |1 + tan |-|| |
              \    \        \2// /
2tan⁡(x2)(−(1−tan⁡2(x2))2(tan⁡2(x2)+1)2+4)(tan⁡2(x2)+1)\frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(- \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 4\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
                 /x\            
            4*cot|-|            
                 \2/            
--------------------------------
              /            2   \
/       2/x\\ |    -1 + cot (x)|
|1 + cot |-||*|7 - ------------|
\        \2// |           2    |
              \    1 + cot (x) /
4cot⁡(x2)(−cot⁡2(x)−1cot⁡2(x)+1+7)(cot⁡2(x2)+1)\frac{4 \cot{\left(\frac{x}{2} \right)}}{\left(- \frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} + 7\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
                  /x\             
             2*tan|-|             
                  \2/             
----------------------------------
              /           2/x\   \
              |      4*tan |-|   |
/       2/x\\ |            \2/   |
|1 + tan |-||*|3 + --------------|
\        \2// |                 2|
              |    /       2/x\\ |
              |    |1 + tan |-|| |
              \    \        \2// /
2tan⁡(x2)(3+4tan⁡2(x2)(tan⁡2(x2)+1)2)(tan⁡2(x2)+1)\frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(3 + \frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
     /    pi\
2*cos|x - --|
     \    2 /
-------------
 7 - cos(2*x)
2cos⁡(x−π2)7−cos⁡(2x)\frac{2 \cos{\left(x - \frac{\pi}{2} \right)}}{7 - \cos{\left(2 x \right)}} 
            1            
-------------------------
/         1      \       
|4 - ------------|*csc(x)
|       2/pi    \|       
|    csc |-- - x||       
\        \2     //
1(4−1csc⁡2(−x+π2))csc⁡(x)\frac{1}{\left(4 - \frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(x \right)}} 
1/((4 - 1/csc(pi/2 - x)^2)*csc(x))
Respuesta numérica [src] 
sin(x)/(4.0 - cos(x)^2)
sin(x)/(4.0 - cos(x)^2)
Potencias [src] 
     /   -I*x    I*x\  
  -I*\- e     + e   /  
-----------------------
  /                  2\
  |    / I*x    -I*x\ |
  |    |e      e    | |
2*|4 - |---- + -----| |
  \    \ 2       2  / /
−i(eix−e−ix)2(4−(eix2+e−ix2)2)- \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(4 - \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2}\right)} 
-i*(-exp(-i*x) + exp(i*x))/(2*(4 - (exp(i*x)/2 + exp(-i*x)/2)^2))
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