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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(5*x)/(1-cos(5*x))

¿Cómo vas a descomponer esta sin(5*x)/(1-cos(5*x)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
  sin(5*x)  
------------
1 - cos(5*x)
sin⁡(5x)1−cos⁡(5x)\frac{\sin{\left(5 x \right)}}{1 - \cos{\left(5 x \right)}} 
sin(5*x)/(1 - cos(5*x))
Simplificación general [src] 
  -sin(5*x)  
-------------
-1 + cos(5*x)
−sin⁡(5x)cos⁡(5x)−1- \frac{\sin{\left(5 x \right)}}{\cos{\left(5 x \right)} - 1} 
-sin(5*x)/(-1 + cos(5*x))
Respuesta numérica [src] 
sin(5*x)/(1.0 - cos(5*x))
sin(5*x)/(1.0 - cos(5*x))
Combinatoria [src] 
  -sin(5*x)  
-------------
-1 + cos(5*x)
−sin⁡(5x)cos⁡(5x)−1- \frac{\sin{\left(5 x \right)}}{\cos{\left(5 x \right)} - 1} 
-sin(5*x)/(-1 + cos(5*x))
Denominador común [src] 
  -sin(5*x)  
-------------
-1 + cos(5*x)
−sin⁡(5x)cos⁡(5x)−1- \frac{\sin{\left(5 x \right)}}{\cos{\left(5 x \right)} - 1} 
-sin(5*x)/(-1 + cos(5*x))
Abrimos la expresión [src] 
                      3                                                                                 5                 
                20*sin (x)                                5*sin(x)                                16*sin (x)              
- -------------------------------------- + -------------------------------------- + --------------------------------------
            5                       3                5                       3                5                       3   
  1 - 16*cos (x) - 5*cos(x) + 20*cos (x)   1 - 16*cos (x) - 5*cos(x) + 20*cos (x)   1 - 16*cos (x) - 5*cos(x) + 20*cos (x)
16sin⁡5(x)−16cos⁡5(x)+20cos⁡3(x)−5cos⁡(x)+1−20sin⁡3(x)−16cos⁡5(x)+20cos⁡3(x)−5cos⁡(x)+1+5sin⁡(x)−16cos⁡5(x)+20cos⁡3(x)−5cos⁡(x)+1\frac{16 \sin^{5}{\left(x \right)}}{- 16 \cos^{5}{\left(x \right)} + 20 \cos^{3}{\left(x \right)} - 5 \cos{\left(x \right)} + 1} - \frac{20 \sin^{3}{\left(x \right)}}{- 16 \cos^{5}{\left(x \right)} + 20 \cos^{3}{\left(x \right)} - 5 \cos{\left(x \right)} + 1} + \frac{5 \sin{\left(x \right)}}{- 16 \cos^{5}{\left(x \right)} + 20 \cos^{3}{\left(x \right)} - 5 \cos{\left(x \right)} + 1} 
-20*sin(x)^3/(1 - 16*cos(x)^5 - 5*cos(x) + 20*cos(x)^3) + 5*sin(x)/(1 - 16*cos(x)^5 - 5*cos(x) + 20*cos(x)^3) + 16*sin(x)^5/(1 - 16*cos(x)^5 - 5*cos(x) + 20*cos(x)^3)
Parte trigonométrica [src] 
     sin(5*x)    
-----------------
       /pi      \
1 - sin|-- + 5*x|
       \2       /
sin⁡(5x)1−sin⁡(5x+π2)\frac{\sin{\left(5 x \right)}}{1 - \sin{\left(5 x + \frac{\pi}{2} \right)}} 
             1              
----------------------------
/       1    \    /      pi\
|1 - --------|*sec|5*x - --|
\    sec(5*x)/    \      2 /
1(1−1sec⁡(5x))sec⁡(5x−π2)\frac{1}{\left(1 - \frac{1}{\sec{\left(5 x \right)}}\right) \sec{\left(5 x - \frac{\pi}{2} \right)}} 
   /      pi\
cos|5*x - --|
   \      2 /
-------------
 1 - cos(5*x)
cos⁡(5x−π2)1−cos⁡(5x)\frac{\cos{\left(5 x - \frac{\pi}{2} \right)}}{1 - \cos{\left(5 x \right)}} 
  -sin(5*x)  
-------------
-1 + cos(5*x)
−sin⁡(5x)cos⁡(5x)−1- \frac{\sin{\left(5 x \right)}}{\cos{\left(5 x \right)} - 1} 
                  /5*x\             
             2*cot|---|             
                  \ 2 /             
------------------------------------
                /            2/5*x\\
                |    -1 + cot |---||
/       2/5*x\\ |             \ 2 /|
|1 + cot |---||*|1 - --------------|
\        \ 2 // |           2/5*x\ |
                |    1 + cot |---| |
                \            \ 2 / /
2cot⁡(5x2)(−cot⁡2(5x2)−1cot⁡2(5x2)+1+1)(cot⁡2(5x2)+1)\frac{2 \cot{\left(\frac{5 x}{2} \right)}}{\left(- \frac{\cot^{2}{\left(\frac{5 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{5 x}{2} \right)} + 1\right)} 
    /      pi\ 
-cos|5*x - --| 
    \      2 / 
---------------
 -1 + cos(5*x)
−cos⁡(5x−π2)cos⁡(5x)−1- \frac{\cos{\left(5 x - \frac{\pi}{2} \right)}}{\cos{\left(5 x \right)} - 1} 
             1              
----------------------------
/          1      \         
|1 - -------------|*csc(5*x)
|       /pi      \|         
|    csc|-- - 5*x||         
\       \2       //
1(1−1csc⁡(−5x+π2))csc⁡(5x)\frac{1}{\left(1 - \frac{1}{\csc{\left(- 5 x + \frac{\pi}{2} \right)}}\right) \csc{\left(5 x \right)}} 
                   /5*x\             
             -2*cot|---|             
                   \ 2 /             
-------------------------------------
                /             2/5*x\\
                |     -1 + cot |---||
/       2/5*x\\ |              \ 2 /|
|1 + cot |---||*|-1 + --------------|
\        \ 2 // |            2/5*x\ |
                |     1 + cot |---| |
                \             \ 2 / /
−2cot⁡(5x2)(cot⁡2(5x2)−1cot⁡2(5x2)+1−1)(cot⁡2(5x2)+1)- \frac{2 \cot{\left(\frac{5 x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{5 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 x}{2} \right)} + 1} - 1\right) \left(\cot^{2}{\left(\frac{5 x}{2} \right)} + 1\right)} 
             -1              
-----------------------------
/           1      \         
|-1 + -------------|*csc(5*x)
|        /pi      \|         
|     csc|-- - 5*x||         
\        \2       //
−1(−1+1csc⁡(−5x+π2))csc⁡(5x)- \frac{1}{\left(-1 + \frac{1}{\csc{\left(- 5 x + \frac{\pi}{2} \right)}}\right) \csc{\left(5 x \right)}} 
                  /5*x\            
             2*tan|---|            
                  \ 2 /            
-----------------------------------
                /           2/5*x\\
                |    1 - tan |---||
/       2/5*x\\ |            \ 2 /|
|1 + tan |---||*|1 - -------------|
\        \ 2 // |           2/5*x\|
                |    1 + tan |---||
                \            \ 2 //
2tan⁡(5x2)(−1−tan⁡2(5x2)tan⁡2(5x2)+1+1)(tan⁡2(5x2)+1)\frac{2 \tan{\left(\frac{5 x}{2} \right)}}{\left(- \frac{1 - \tan^{2}{\left(\frac{5 x}{2} \right)}}{\tan^{2}{\left(\frac{5 x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{5 x}{2} \right)} + 1\right)} 
    -sin(5*x)     
------------------
        /pi      \
-1 + sin|-- + 5*x|
        \2       /
−sin⁡(5x)sin⁡(5x+π2)−1- \frac{\sin{\left(5 x \right)}}{\sin{\left(5 x + \frac{\pi}{2} \right)} - 1} 
             -1              
-----------------------------
/        1    \    /      pi\
|-1 + --------|*sec|5*x - --|
\     sec(5*x)/    \      2 /
−1(−1+1sec⁡(5x))sec⁡(5x−π2)- \frac{1}{\left(-1 + \frac{1}{\sec{\left(5 x \right)}}\right) \sec{\left(5 x - \frac{\pi}{2} \right)}} 
                  /5*x\             
            -2*tan|---|             
                  \ 2 /             
------------------------------------
                /            2/5*x\\
                |     1 - tan |---||
/       2/5*x\\ |             \ 2 /|
|1 + tan |---||*|-1 + -------------|
\        \ 2 // |            2/5*x\|
                |     1 + tan |---||
                \             \ 2 //
−2tan⁡(5x2)(1−tan⁡2(5x2)tan⁡2(5x2)+1−1)(tan⁡2(5x2)+1)- \frac{2 \tan{\left(\frac{5 x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{5 x}{2} \right)}}{\tan^{2}{\left(\frac{5 x}{2} \right)} + 1} - 1\right) \left(\tan^{2}{\left(\frac{5 x}{2} \right)} + 1\right)} 
           1           
-----------------------
/       1    \         
|1 - --------|*csc(5*x)
\    sec(5*x)/
1(1−1sec⁡(5x))csc⁡(5x)\frac{1}{\left(1 - \frac{1}{\sec{\left(5 x \right)}}\right) \csc{\left(5 x \right)}} 
1/((1 - 1/sec(5*x))*csc(5*x))
Potencias [src] 
   /   -5*I*x    5*I*x\ 
-I*\- e       + e     / 
------------------------
  /     -5*I*x    5*I*x\
  |    e         e     |
2*|1 - ------- - ------|
  \       2        2   /
−i(e5ix−e−5ix)2(−e5ix2+1−e−5ix2)- \frac{i \left(e^{5 i x} - e^{- 5 i x}\right)}{2 \left(- \frac{e^{5 i x}}{2} + 1 - \frac{e^{- 5 i x}}{2}\right)} 
-i*(-exp(-5*i*x) + exp(5*i*x))/(2*(1 - exp(-5*i*x)/2 - exp(5*i*x)/2))
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