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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(sqrt(x))/(2*sqrt(x)*(1+sin(sqrt(x))))-cos(sqrt(x))/(2*sqrt(x)*(sin(sqrt(x))-1))

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

Expresión a simplificar:

Solución

Ha introducido [src] 
          /  ___\                    /  ___\       
       cos\\/ x /                 cos\\/ x /       
------------------------ - ------------------------
    ___ /       /  ___\\       ___ /   /  ___\    \
2*\/ x *\1 + sin\\/ x //   2*\/ x *\sin\\/ x / - 1/
cos⁡(x)2x(sin⁡(x)+1)−cos⁡(x)2x(sin⁡(x)−1)\frac{\cos{\left(\sqrt{x} \right)}}{2 \sqrt{x} \left(\sin{\left(\sqrt{x} \right)} + 1\right)} - \frac{\cos{\left(\sqrt{x} \right)}}{2 \sqrt{x} \left(\sin{\left(\sqrt{x} \right)} - 1\right)} 
cos(sqrt(x))/(((2*sqrt(x))*(1 + sin(sqrt(x))))) - cos(sqrt(x))/((2*sqrt(x))*(sin(sqrt(x)) - 1))
Simplificación general [src] 
       1        
----------------
  ___    /  ___\
\/ x *cos\\/ x /
1xcos⁡(x)\frac{1}{\sqrt{x} \cos{\left(\sqrt{x} \right)}} 
1/(sqrt(x)*cos(sqrt(x)))
Respuesta numérica [src] 
0.5*x^(-0.5)*cos(sqrt(x))/(1.0 + sin(sqrt(x))) - 0.5*x^(-0.5)*cos(sqrt(x))/(-1.0 + sin(sqrt(x)))
0.5*x^(-0.5)*cos(sqrt(x))/(1.0 + sin(sqrt(x))) - 0.5*x^(-0.5)*cos(sqrt(x))/(-1.0 + sin(sqrt(x)))
Combinatoria [src] 
                  /  ___\               
              -cos\\/ x /               
----------------------------------------
  ___ /       /  ___\\ /        /  ___\\
\/ x *\1 + sin\\/ x //*\-1 + sin\\/ x //
−cos⁡(x)x(sin⁡(x)−1)(sin⁡(x)+1)- \frac{\cos{\left(\sqrt{x} \right)}}{\sqrt{x} \left(\sin{\left(\sqrt{x} \right)} - 1\right) \left(\sin{\left(\sqrt{x} \right)} + 1\right)} 
-cos(sqrt(x))/(sqrt(x)*(1 + sin(sqrt(x)))*(-1 + sin(sqrt(x))))
Denominador racional [src] 
      ___ /       /  ___\\    /  ___\       ___ /        /  ___\\    /  ___\
- 2*\/ x *\1 + sin\\/ x //*cos\\/ x / + 2*\/ x *\-1 + sin\\/ x //*cos\\/ x /
----------------------------------------------------------------------------
                       /       /  ___\\ /        /  ___\\                   
                   4*x*\1 + sin\\/ x //*\-1 + sin\\/ x //
2x(sin⁡(x)−1)cos⁡(x)−2x(sin⁡(x)+1)cos⁡(x)4x(sin⁡(x)−1)(sin⁡(x)+1)\frac{2 \sqrt{x} \left(\sin{\left(\sqrt{x} \right)} - 1\right) \cos{\left(\sqrt{x} \right)} - 2 \sqrt{x} \left(\sin{\left(\sqrt{x} \right)} + 1\right) \cos{\left(\sqrt{x} \right)}}{4 x \left(\sin{\left(\sqrt{x} \right)} - 1\right) \left(\sin{\left(\sqrt{x} \right)} + 1\right)} 
(-2*sqrt(x)*(1 + sin(sqrt(x)))*cos(sqrt(x)) + 2*sqrt(x)*(-1 + sin(sqrt(x)))*cos(sqrt(x)))/(4*x*(1 + sin(sqrt(x)))*(-1 + sin(sqrt(x))))
Unión de expresiones racionales [src] 
                  /  ___\               
              -cos\\/ x /               
----------------------------------------
  ___ /       /  ___\\ /        /  ___\\
\/ x *\1 + sin\\/ x //*\-1 + sin\\/ x //
−cos⁡(x)x(sin⁡(x)−1)(sin⁡(x)+1)- \frac{\cos{\left(\sqrt{x} \right)}}{\sqrt{x} \left(\sin{\left(\sqrt{x} \right)} - 1\right) \left(\sin{\left(\sqrt{x} \right)} + 1\right)} 
-cos(sqrt(x))/(sqrt(x)*(1 + sin(sqrt(x)))*(-1 + sin(sqrt(x))))
Potencias [src] 
          /  ___\                     /  ___\       
       cos\\/ x /                  cos\\/ x /       
------------------------ - -------------------------
  ___ /         /  ___\\     ___ /          /  ___\\
\/ x *\2 + 2*sin\\/ x //   \/ x *\-2 + 2*sin\\/ x //
cos⁡(x)x(2sin⁡(x)+2)−cos⁡(x)x(2sin⁡(x)−2)\frac{\cos{\left(\sqrt{x} \right)}}{\sqrt{x} \left(2 \sin{\left(\sqrt{x} \right)} + 2\right)} - \frac{\cos{\left(\sqrt{x} \right)}}{\sqrt{x} \left(2 \sin{\left(\sqrt{x} \right)} - 2\right)} 
               ___         ___                             ___         ___          
           I*\/ x     -I*\/ x                          I*\/ x     -I*\/ x           
          e          e                                e          e                  
          -------- + ---------                        -------- + ---------          
             2           2                               2           2              
---------------------------------------- - -----------------------------------------
        /      /        ___        ___\\           /       /        ___        ___\\
        |      |   -I*\/ x     I*\/ x ||           |       |   -I*\/ x     I*\/ x ||
    ___ |    I*\- e         + e       /|       ___ |     I*\- e         + e       /|
2*\/ x *|1 - --------------------------|   2*\/ x *|-1 - --------------------------|
        \                2             /           \                 2             /
eix2+e−ix22x(−i(eix−e−ix)2+1)−eix2+e−ix22x(−i(eix−e−ix)2−1)\frac{\frac{e^{i \sqrt{x}}}{2} + \frac{e^{- i \sqrt{x}}}{2}}{2 \sqrt{x} \left(- \frac{i \left(e^{i \sqrt{x}} - e^{- i \sqrt{x}}\right)}{2} + 1\right)} - \frac{\frac{e^{i \sqrt{x}}}{2} + \frac{e^{- i \sqrt{x}}}{2}}{2 \sqrt{x} \left(- \frac{i \left(e^{i \sqrt{x}} - e^{- i \sqrt{x}}\right)}{2} - 1\right)} 
(exp(i*sqrt(x))/2 + exp(-i*sqrt(x))/2)/(2*sqrt(x)*(1 - i*(-exp(-i*sqrt(x)) + exp(i*sqrt(x)))/2)) - (exp(i*sqrt(x))/2 + exp(-i*sqrt(x))/2)/(2*sqrt(x)*(-1 - i*(-exp(-i*sqrt(x)) + exp(i*sqrt(x)))/2))
Abrimos la expresión [src] 
            /  ___\                         /  ___\          
         cos\\/ x /                      cos\\/ x /          
---------------------------- - ------------------------------
    ___       ___    /  ___\         ___       ___    /  ___\
2*\/ x  + 2*\/ x *sin\\/ x /   - 2*\/ x  + 2*\/ x *sin\\/ x /
cos⁡(x)2xsin⁡(x)+2x−cos⁡(x)2xsin⁡(x)−2x\frac{\cos{\left(\sqrt{x} \right)}}{2 \sqrt{x} \sin{\left(\sqrt{x} \right)} + 2 \sqrt{x}} - \frac{\cos{\left(\sqrt{x} \right)}}{2 \sqrt{x} \sin{\left(\sqrt{x} \right)} - 2 \sqrt{x}} 
cos(sqrt(x))/(2*sqrt(x) + 2*sqrt(x)*sin(sqrt(x))) - cos(sqrt(x))/(-2*sqrt(x) + 2*sqrt(x)*sin(sqrt(x)))
Parte trigonométrica [src] 
                      2                      
---------------------------------------------
  ___ /            1        \    /pi     ___\
\/ x *|1 + -----------------|*csc|-- - \/ x |
      |       /pi       ___\|    \2         /
      |    csc|-- - 2*\/ x ||                
      \       \2           //
2x(1+1csc⁡(−2x+π2))csc⁡(−x+π2)\frac{2}{\sqrt{x} \left(1 + \frac{1}{\csc{\left(- 2 \sqrt{x} + \frac{\pi}{2} \right)}}\right) \csc{\left(- \sqrt{x} + \frac{\pi}{2} \right)}} 
   /  ___\
sec\\/ x /
----------
    ___   
  \/ x
sec⁡(x)x\frac{\sec{\left(\sqrt{x} \right)}}{\sqrt{x}} 
                         /  ___\                                           /  ___\                
                        2|\/ x |                                          2|\/ x |                
                -1 + cot |-----|                                  -1 + cot |-----|                
                         \  2  /                                           \  2  /                
----------------------------------------------- - ------------------------------------------------
                          /           /  ___\ \                             /            /  ___\ \
                          |           |\/ x | |                             |            |\/ x | |
        /        /  ___\\ |      2*cot|-----| |           /        /  ___\\ |       2*cot|-----| |
    ___ |       2|\/ x || |           \  2  / |       ___ |       2|\/ x || |            \  2  / |
2*\/ x *|1 + cot |-----||*|1 + ---------------|   2*\/ x *|1 + cot |-----||*|-1 + ---------------|
        \        \  2  // |            /  ___\|           \        \  2  // |             /  ___\|
                          |           2|\/ x ||                             |            2|\/ x ||
                          |    1 + cot |-----||                             |     1 + cot |-----||
                          \            \  2  //                             \             \  2  //
cot⁡2(x2)−12x(1+2cot⁡(x2)cot⁡2(x2)+1)(cot⁡2(x2)+1)−cot⁡2(x2)−12x(−1+2cot⁡(x2)cot⁡2(x2)+1)(cot⁡2(x2)+1)\frac{\cot^{2}{\left(\frac{\sqrt{x}}{2} \right)} - 1}{2 \sqrt{x} \left(1 + \frac{2 \cot{\left(\frac{\sqrt{x}}{2} \right)}}{\cot^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1\right)} - \frac{\cot^{2}{\left(\frac{\sqrt{x}}{2} \right)} - 1}{2 \sqrt{x} \left(-1 + \frac{2 \cot{\left(\frac{\sqrt{x}}{2} \right)}}{\cot^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1\right)} 
                   1                                           1                    
---------------------------------------- - -----------------------------------------
    ___ /           1       \    /  ___\       ___ /            1       \    /  ___\
2*\/ x *|1 + ---------------|*sec\\/ x /   2*\/ x *|-1 + ---------------|*sec\\/ x /
        |       /  ___   pi\|                      |        /  ___   pi\|           
        |    sec|\/ x  - --||                      |     sec|\/ x  - --||           
        \       \        2 //                      \        \        2 //
12x(1+1sec⁡(x−π2))sec⁡(x)−12x(−1+1sec⁡(x−π2))sec⁡(x)\frac{1}{2 \sqrt{x} \left(1 + \frac{1}{\sec{\left(\sqrt{x} - \frac{\pi}{2} \right)}}\right) \sec{\left(\sqrt{x} \right)}} - \frac{1}{2 \sqrt{x} \left(-1 + \frac{1}{\sec{\left(\sqrt{x} - \frac{\pi}{2} \right)}}\right) \sec{\left(\sqrt{x} \right)}} 
           /  ___\      
      2*cos\\/ x /      
------------------------
  ___ /       /    ___\\
\/ x *\1 + cos\2*\/ x //
2cos⁡(x)x(cos⁡(2x)+1)\frac{2 \cos{\left(\sqrt{x} \right)}}{\sqrt{x} \left(\cos{\left(2 \sqrt{x} \right)} + 1\right)} 
       /  ___   pi\                /  ___   pi\     
    sin|\/ x  + --|             sin|\/ x  + --|     
       \        2 /                \        2 /     
------------------------ - -------------------------
    ___ /       /  ___\\       ___ /        /  ___\\
2*\/ x *\1 + sin\\/ x //   2*\/ x *\-1 + sin\\/ x //
sin⁡(x+π2)2x(sin⁡(x)+1)−sin⁡(x+π2)2x(sin⁡(x)−1)\frac{\sin{\left(\sqrt{x} + \frac{\pi}{2} \right)}}{2 \sqrt{x} \left(\sin{\left(\sqrt{x} \right)} + 1\right)} - \frac{\sin{\left(\sqrt{x} + \frac{\pi}{2} \right)}}{2 \sqrt{x} \left(\sin{\left(\sqrt{x} \right)} - 1\right)} 
                        /  ___\                                           /  ___\                 
                       2|\/ x |                                          2|\/ x |                 
                1 - tan |-----|                                   1 - tan |-----|                 
                        \  2  /                                           \  2  /                 
----------------------------------------------- - ------------------------------------------------
                          /           /  ___\ \                             /            /  ___\ \
                          |           |\/ x | |                             |            |\/ x | |
        /        /  ___\\ |      2*tan|-----| |           /        /  ___\\ |       2*tan|-----| |
    ___ |       2|\/ x || |           \  2  / |       ___ |       2|\/ x || |            \  2  / |
2*\/ x *|1 + tan |-----||*|1 + ---------------|   2*\/ x *|1 + tan |-----||*|-1 + ---------------|
        \        \  2  // |            /  ___\|           \        \  2  // |             /  ___\|
                          |           2|\/ x ||                             |            2|\/ x ||
                          |    1 + tan |-----||                             |     1 + tan |-----||
                          \            \  2  //                             \             \  2  //
1−tan⁡2(x2)2x(1+2tan⁡(x2)tan⁡2(x2)+1)(tan⁡2(x2)+1)−1−tan⁡2(x2)2x(−1+2tan⁡(x2)tan⁡2(x2)+1)(tan⁡2(x2)+1)\frac{1 - \tan^{2}{\left(\frac{\sqrt{x}}{2} \right)}}{2 \sqrt{x} \left(1 + \frac{2 \tan{\left(\frac{\sqrt{x}}{2} \right)}}{\tan^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1\right)} - \frac{1 - \tan^{2}{\left(\frac{\sqrt{x}}{2} \right)}}{2 \sqrt{x} \left(-1 + \frac{2 \tan{\left(\frac{\sqrt{x}}{2} \right)}}{\tan^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1\right)} 
                 1                                     1                  
----------------------------------- - ------------------------------------
    ___ /        1     \    /  ___\       ___ /         1     \    /  ___\
2*\/ x *|1 + ----------|*sec\\/ x /   2*\/ x *|-1 + ----------|*sec\\/ x /
        |       /  ___\|                      |        /  ___\|           
        \    csc\\/ x //                      \     csc\\/ x //
12x(1+1csc⁡(x))sec⁡(x)−12x(−1+1csc⁡(x))sec⁡(x)\frac{1}{2 \sqrt{x} \left(1 + \frac{1}{\csc{\left(\sqrt{x} \right)}}\right) \sec{\left(\sqrt{x} \right)}} - \frac{1}{2 \sqrt{x} \left(-1 + \frac{1}{\csc{\left(\sqrt{x} \right)}}\right) \sec{\left(\sqrt{x} \right)}} 
               /        /  ___\\             
               |       2|\/ x ||             
             2*|1 - tan |-----||             
               \        \  2  //             
---------------------------------------------
      /        /  ___\\ /           2/  ___\\
  ___ |       2|\/ x || |    1 - tan \\/ x /|
\/ x *|1 + tan |-----||*|1 + ---------------|
      \        \  2  // |           2/  ___\|
                        \    1 + tan \\/ x //
2(1−tan⁡2(x2))x(1−tan⁡2(x)tan⁡2(x)+1+1)(tan⁡2(x2)+1)\frac{2 \left(1 - \tan^{2}{\left(\frac{\sqrt{x}}{2} \right)}\right)}{\sqrt{x} \left(\frac{1 - \tan^{2}{\left(\sqrt{x} \right)}}{\tan^{2}{\left(\sqrt{x} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1\right)} 
             /  ___\                         /  ___\          
          cos\\/ x /                      cos\\/ x /          
----------------------------- - ------------------------------
    ___ /       /  ___   pi\\       ___ /        /  ___   pi\\
2*\/ x *|1 + cos|\/ x  - --||   2*\/ x *|-1 + cos|\/ x  - --||
        \       \        2 //           \        \        2 //
cos⁡(x)2x(cos⁡(x−π2)+1)−cos⁡(x)2x(cos⁡(x−π2)−1)\frac{\cos{\left(\sqrt{x} \right)}}{2 \sqrt{x} \left(\cos{\left(\sqrt{x} - \frac{\pi}{2} \right)} + 1\right)} - \frac{\cos{\left(\sqrt{x} \right)}}{2 \sqrt{x} \left(\cos{\left(\sqrt{x} - \frac{\pi}{2} \right)} - 1\right)} 
                 2                 
-----------------------------------
  ___ /         1      \    /  ___\
\/ x *|1 + ------------|*sec\\/ x /
      |       /    ___\|           
      \    sec\2*\/ x //
2x(1+1sec⁡(2x))sec⁡(x)\frac{2}{\sqrt{x} \left(1 + \frac{1}{\sec{\left(2 \sqrt{x} \right)}}\right) \sec{\left(\sqrt{x} \right)}} 
            /  ___\     
           2|\/ x |     
    1 + cot |-----|     
            \  2  /     
------------------------
      /         /  ___\\
  ___ |        2|\/ x ||
\/ x *|-1 + cot |-----||
      \         \  2  //
cot⁡2(x2)+1x(cot⁡2(x2)−1)\frac{\cot^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1}{\sqrt{x} \left(\cot^{2}{\left(\frac{\sqrt{x}}{2} \right)} - 1\right)} 
                   1                                           1                    
---------------------------------------- - -----------------------------------------
    ___ /        1     \    /pi     ___\       ___ /         1     \    /pi     ___\
2*\/ x *|1 + ----------|*csc|-- - \/ x |   2*\/ x *|-1 + ----------|*csc|-- - \/ x |
        |       /  ___\|    \2         /           |        /  ___\|    \2         /
        \    csc\\/ x //                           \     csc\\/ x //
12x(1+1csc⁡(x))csc⁡(−x+π2)−12x(−1+1csc⁡(x))csc⁡(−x+π2)\frac{1}{2 \sqrt{x} \left(1 + \frac{1}{\csc{\left(\sqrt{x} \right)}}\right) \csc{\left(- \sqrt{x} + \frac{\pi}{2} \right)}} - \frac{1}{2 \sqrt{x} \left(-1 + \frac{1}{\csc{\left(\sqrt{x} \right)}}\right) \csc{\left(- \sqrt{x} + \frac{\pi}{2} \right)}} 
          /  ___\
         2|\/ x |
1 - 2*sin |-----|
          \  2  /
-----------------
  ___    2/  ___\
\/ x *cos \\/ x /
1−2sin⁡2(x2)xcos⁡2(x)\frac{1 - 2 \sin^{2}{\left(\frac{\sqrt{x}}{2} \right)}}{\sqrt{x} \cos^{2}{\left(\sqrt{x} \right)}} 
       1        
----------------
  ___    /  ___\
\/ x *cos\\/ x /
1xcos⁡(x)\frac{1}{\sqrt{x} \cos{\left(\sqrt{x} \right)}} 
            /  ___\    
           2|\/ x |    
    1 + tan |-----|    
            \  2  /    
-----------------------
      /        /  ___\\
  ___ |       2|\/ x ||
\/ x *|1 - tan |-----||
      \        \  2  //
tan⁡2(x2)+1x(1−tan⁡2(x2))\frac{\tan^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1}{\sqrt{x} \left(1 - \tan^{2}{\left(\frac{\sqrt{x}}{2} \right)}\right)} 
          1          
---------------------
  ___    /  ___   pi\
\/ x *sin|\/ x  + --|
         \        2 /
1xsin⁡(x+π2)\frac{1}{\sqrt{x} \sin{\left(\sqrt{x} + \frac{\pi}{2} \right)}} 
   /pi     ___\
csc|-- - \/ x |
   \2         /
---------------
       ___     
     \/ x
csc⁡(−x+π2)x\frac{\csc{\left(- \sqrt{x} + \frac{\pi}{2} \right)}}{\sqrt{x}} 
               /         /  ___\\             
               |        2|\/ x ||             
             2*|-1 + cot |-----||             
               \         \  2  //             
----------------------------------------------
      /        /  ___\\ /            2/  ___\\
  ___ |       2|\/ x || |    -1 + cot \\/ x /|
\/ x *|1 + cot |-----||*|1 + ----------------|
      \        \  2  // |           2/  ___\ |
                        \    1 + cot \\/ x / /
2(cot⁡2(x2)−1)x(cot⁡2(x)−1cot⁡2(x)+1+1)(cot⁡2(x2)+1)\frac{2 \left(\cot^{2}{\left(\frac{\sqrt{x}}{2} \right)} - 1\right)}{\sqrt{x} \left(\frac{\cot^{2}{\left(\sqrt{x} \right)} - 1}{\cot^{2}{\left(\sqrt{x} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1\right)} 
           /  ___   pi\      
      2*sin|\/ x  + --|      
           \        2 /      
-----------------------------
  ___ /       /pi       ___\\
\/ x *|1 + sin|-- + 2*\/ x ||
      \       \2           //
2sin⁡(x+π2)x(sin⁡(2x+π2)+1)\frac{2 \sin{\left(\sqrt{x} + \frac{\pi}{2} \right)}}{\sqrt{x} \left(\sin{\left(2 \sqrt{x} + \frac{\pi}{2} \right)} + 1\right)} 
2*sin(sqrt(x) + pi/2)/(sqrt(x)*(1 + sin(pi/2 + 2*sqrt(x))))
Denominador común [src] 
            /  ___\        
        -cos\\/ x /        
---------------------------
    ___     ___    2/  ___\
- \/ x  + \/ x *sin \\/ x /
−cos⁡(x)xsin⁡2(x)−x- \frac{\cos{\left(\sqrt{x} \right)}}{\sqrt{x} \sin^{2}{\left(\sqrt{x} \right)} - \sqrt{x}} 
-cos(sqrt(x))/(-sqrt(x) + sqrt(x)*sin(sqrt(x))^2)
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