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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(x)^(2/x)*(-2*log(cos(x))/x^2-2*sin(x)/(x*cos(x)))

¿Cómo vas a descomponer esta cos(x)^(2/x)*(-2*log(cos(x))/x^2-2*sin(x)/(x*cos(x))) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
        2                            
        -                            
        x /-2*log(cos(x))   2*sin(x)\
(cos(x)) *|-------------- - --------|
          |       2         x*cos(x)|
          \      x                  /
(−2sin⁡(x)xcos⁡(x)+(−1)2log⁡(cos⁡(x))x2)cos⁡2x(x)\left(- \frac{2 \sin{\left(x \right)}}{x \cos{\left(x \right)}} + \frac{\left(-1\right) 2 \log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) \cos^{\frac{2}{x}}{\left(x \right)} 
cos(x)^(2/x)*((-2*log(cos(x)))/x^2 - 2*sin(x)/(x*cos(x)))
Simplificación general [src] 
                2                                
           -1 + -                                
                x                                
-2*(cos(x))      *(x*sin(x) + cos(x)*log(cos(x)))
-------------------------------------------------
                         2                       
                        x
−2(xsin⁡(x)+log⁡(cos⁡(x))cos⁡(x))cos⁡−1+2x(x)x2- \frac{2 \left(x \sin{\left(x \right)} + \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{-1 + \frac{2}{x}}{\left(x \right)}}{x^{2}} 
-2*cos(x)^(-1 + 2/x)*(x*sin(x) + cos(x)*log(cos(x)))/x^2
Respuesta numérica [src] 
cos(x)^(2.0/x)*(-2.0*log(cos(x))/x^2 - 2.0*sin(x)/(x*cos(x)))
cos(x)^(2.0/x)*(-2.0*log(cos(x))/x^2 - 2.0*sin(x)/(x*cos(x)))
Potencias [src] 
   _________                             
x /    2     /  2*log(cos(x))   2*sin(x)\
\/  cos (x) *|- ------------- - --------|
             |         2        x*cos(x)|
             \        x                 /
(−2sin⁡(x)xcos⁡(x)−2log⁡(cos⁡(x))x2)(cos⁡2(x))1x\left(- \frac{2 \sin{\left(x \right)}}{x \cos{\left(x \right)}} - \frac{2 \log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) \left(\cos^{2}{\left(x \right)}\right)^{\frac{1}{x}} 
              2                                             
              - /       / I*x    -I*x\                     \
              x |       |e      e    |                     |
/ I*x    -I*x\  |  2*log|---- + -----|     /   -I*x    I*x\|
|e      e    |  |       \ 2       2  /   I*\- e     + e   /|
|---- + -----| *|- ------------------- + ------------------|
\ 2       2  /  |            2              / I*x    -I*x\ |
                |           x               |e      e    | |
                |                         x*|---- + -----| |
                \                           \ 2       2  / /
(i(eix−e−ix)x(eix2+e−ix2)−2log⁡(eix2+e−ix2)x2)(eix2+e−ix2)2x\left(\frac{i \left(e^{i x} - e^{- i x}\right)}{x \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)} - \frac{2 \log{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2} \right)}}{x^{2}}\right) \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{\frac{2}{x}} 
        2                             
        -                             
        x /  2*log(cos(x))   2*sin(x)\
(cos(x)) *|- ------------- - --------|
          |         2        x*cos(x)|
          \        x                 /
(−2sin⁡(x)xcos⁡(x)−2log⁡(cos⁡(x))x2)cos⁡2x(x)\left(- \frac{2 \sin{\left(x \right)}}{x \cos{\left(x \right)}} - \frac{2 \log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) \cos^{\frac{2}{x}}{\left(x \right)} 
cos(x)^(2/x)*(-2*log(cos(x))/x^2 - 2*sin(x)/(x*cos(x)))
Denominador común [src] 
 /            2                    2                   \ 
 |            -                    -                   | 
 |            x                    x                   | 
-\2*x*(cos(x)) *sin(x) + 2*(cos(x)) *cos(x)*log(cos(x))/ 
---------------------------------------------------------
                         2                               
                        x *cos(x)
−2xsin⁡(x)cos⁡2x(x)+2log⁡(cos⁡(x))cos⁡(x)cos⁡2x(x)x2cos⁡(x)- \frac{2 x \sin{\left(x \right)} \cos^{\frac{2}{x}}{\left(x \right)} + 2 \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} \cos^{\frac{2}{x}}{\left(x \right)}}{x^{2} \cos{\left(x \right)}} 
-(2*x*cos(x)^(2/x)*sin(x) + 2*cos(x)^(2/x)*cos(x)*log(cos(x)))/(x^2*cos(x))
Combinatoria [src] 
           2                                
           -                                
           x                                
-2*(cos(x)) *(x*sin(x) + cos(x)*log(cos(x)))
--------------------------------------------
                  2                         
                 x *cos(x)
−2(xsin⁡(x)+log⁡(cos⁡(x))cos⁡(x))cos⁡2x(x)x2cos⁡(x)- \frac{2 \left(x \sin{\left(x \right)} + \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{\frac{2}{x}}{\left(x \right)}}{x^{2} \cos{\left(x \right)}} 
-2*cos(x)^(2/x)*(x*sin(x) + cos(x)*log(cos(x)))/(x^2*cos(x))
Unión de expresiones racionales [src] 
          2                                 
          -                                 
          x                                 
2*(cos(x)) *(-x*sin(x) - cos(x)*log(cos(x)))
--------------------------------------------
                  2                         
                 x *cos(x)
2(−xsin⁡(x)−log⁡(cos⁡(x))cos⁡(x))cos⁡2x(x)x2cos⁡(x)\frac{2 \left(- x \sin{\left(x \right)} - \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{\frac{2}{x}}{\left(x \right)}}{x^{2} \cos{\left(x \right)}} 
2*cos(x)^(2/x)*(-x*sin(x) - cos(x)*log(cos(x)))/(x^2*cos(x))
Abrimos la expresión [src] 
        2                            
        -                            
        x /-2*log(cos(x))   2*sin(x)\
(cos(x)) *|-------------- - --------|
          |       2         x*cos(x)|
          \      x                  /
((−1)2log⁡(cos⁡(x))x2−2sin⁡(x)xcos⁡(x))cos⁡2x(x)\left(\frac{\left(-1\right) 2 \log{\left(\cos{\left(x \right)} \right)}}{x^{2}} - \frac{2 \sin{\left(x \right)}}{x \cos{\left(x \right)}}\right) \cos^{\frac{2}{x}}{\left(x \right)} 
            2                         2       
            -                         -       
            x                         x       
  2*(cos(x)) *log(cos(x))   2*(cos(x)) *sin(x)
- ----------------------- - ------------------
              2                  x*cos(x)     
             x
−2sin⁡(x)cos⁡2x(x)xcos⁡(x)−2log⁡(cos⁡(x))cos⁡2x(x)x2- \frac{2 \sin{\left(x \right)} \cos^{\frac{2}{x}}{\left(x \right)}}{x \cos{\left(x \right)}} - \frac{2 \log{\left(\cos{\left(x \right)} \right)} \cos^{\frac{2}{x}}{\left(x \right)}}{x^{2}} 
-2*cos(x)^(2/x)*log(cos(x))/x^2 - 2*cos(x)^(2/x)*sin(x)/(x*cos(x))
Parte trigonométrica [src] 
                  /   /     1     \              \
                2 |log|-----------|              |
                - |   |   /pi    \|      /pi    \|
                x |   |csc|-- - x||   csc|-- - x||
   /     1     \  |   \   \2     //      \2     /|
-2*|-----------| *|---------------- + -----------|
   |   /pi    \|  \       x              csc(x)  /
   |csc|-- - x||                                  
   \   \2     //                                  
--------------------------------------------------
                        x
−2(csc⁡(−x+π2)csc⁡(x)+log⁡(1csc⁡(−x+π2))x)(1csc⁡(−x+π2))2xx- \frac{2 \left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} + \frac{\log{\left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{x}\right) \left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{\frac{2}{x}}}{x} 
           2                       
           -                       
           x /log(cos(x))         \
-2*(cos(x)) *|----------- + tan(x)|
             \     x              /
-----------------------------------
                 x
−2(tan⁡(x)+log⁡(cos⁡(x))x)cos⁡2x(x)x- \frac{2 \left(\tan{\left(x \right)} + \frac{\log{\left(\cos{\left(x \right)} \right)}}{x}\right) \cos^{\frac{2}{x}}{\left(x \right)}}{x} 
        2                                  
        - /       /  1   \                \
        x |  2*log|------|                |
/  1   \  |       \sec(x)/      2*sec(x)  |
|------| *|- ------------- - -------------|
\sec(x)/  |         2             /    pi\|
          |        x         x*sec|x - --||
          \                       \    2 //
(−2sec⁡(x)xsec⁡(x−π2)−2log⁡(1sec⁡(x))x2)(1sec⁡(x))2x\left(- \frac{2 \sec{\left(x \right)}}{x \sec{\left(x - \frac{\pi}{2} \right)}} - \frac{2 \log{\left(\frac{1}{\sec{\left(x \right)}} \right)}}{x^{2}}\right) \left(\frac{1}{\sec{\left(x \right)}}\right)^{\frac{2}{x}} 
        2 /                       /    pi\\
        - |                  2*cos|x - --||
        x |  2*log(cos(x))        \    2 /|
(cos(x)) *|- ------------- - -------------|
          |         2           x*cos(x)  |
          \        x                      /
(−2cos⁡(x−π2)xcos⁡(x)−2log⁡(cos⁡(x))x2)cos⁡2x(x)\left(- \frac{2 \cos{\left(x - \frac{\pi}{2} \right)}}{x \cos{\left(x \right)}} - \frac{2 \log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) \cos^{\frac{2}{x}}{\left(x \right)} 
               /       /     1     \                \
             2 |  2*log|-----------|                |
             - |       |   /pi    \|        /pi    \|
             x |       |csc|-- - x||   2*csc|-- - x||
/     1     \  |       \   \2     //        \2     /|
|-----------| *|- ------------------ - -------------|
|   /pi    \|  |           2              x*csc(x)  |
|csc|-- - x||  \          x                         /
\   \2     //
(−2csc⁡(−x+π2)xcsc⁡(x)−2log⁡(1csc⁡(−x+π2))x2)(1csc⁡(−x+π2))2x\left(- \frac{2 \csc{\left(- x + \frac{\pi}{2} \right)}}{x \csc{\left(x \right)}} - \frac{2 \log{\left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{x^{2}}\right) \left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{\frac{2}{x}} 
        2                             
        - /       /  1   \           \
        x |  2*log|------|           |
/  1   \  |       \sec(x)/   2*sec(x)|
|------| *|- ------------- - --------|
\sec(x)/  |         2        x*csc(x)|
          \        x                 /
(−2sec⁡(x)xcsc⁡(x)−2log⁡(1sec⁡(x))x2)(1sec⁡(x))2x\left(- \frac{2 \sec{\left(x \right)}}{x \csc{\left(x \right)}} - \frac{2 \log{\left(\frac{1}{\sec{\left(x \right)}} \right)}}{x^{2}}\right) \left(\frac{1}{\sec{\left(x \right)}}\right)^{\frac{2}{x}} 
                   /            /        2/x\\\
                 2 |            |-1 + cot |-|||
                 - |            |         \2/||
                 x |         log|------------||
   /        2/x\\  |            |       2/x\ ||
   |-1 + cot |-||  |            |1 + cot |-| ||
   |         \2/|  |  1         \        \2/ /|
-2*|------------| *|------ + -----------------|
   |       2/x\ |  \cot(x)           x        /
   |1 + cot |-| |                              
   \        \2/ /                              
-----------------------------------------------
                       x
−2(cot⁡2(x2)−1cot⁡2(x2)+1)2x(1cot⁡(x)+log⁡(cot⁡2(x2)−1cot⁡2(x2)+1)x)x- \frac{2 \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{2}{x}} \left(\frac{1}{\cot{\left(x \right)}} + \frac{\log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x}\right)}{x} 
           2                            
           - /   /  1   \              \
           x |log|------|              |
   /  1   \  |   \sec(x)/      sec(x)  |
-2*|------| *|----------- + -----------|
   \sec(x)/  |     x           /    pi\|
             |              sec|x - --||
             \                 \    2 //
----------------------------------------
                   x
−2(sec⁡(x)sec⁡(x−π2)+log⁡(1sec⁡(x))x)(1sec⁡(x))2xx- \frac{2 \left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{\log{\left(\frac{1}{\sec{\left(x \right)}} \right)}}{x}\right) \left(\frac{1}{\sec{\left(x \right)}}\right)^{\frac{2}{x}}}{x} 
             2                                       
             - /       /   /    pi\\                \
             x |  2*log|sin|x + --||                |
/   /    pi\\  |       \   \    2 //      2*sin(x)  |
|sin|x + --|| *|- ------------------ - -------------|
\   \    2 //  |           2                /    pi\|
               |          x            x*sin|x + --||
               \                            \    2 //
(−2sin⁡(x)xsin⁡(x+π2)−2log⁡(sin⁡(x+π2))x2)sin⁡2x(x+π2)\left(- \frac{2 \sin{\left(x \right)}}{x \sin{\left(x + \frac{\pi}{2} \right)}} - \frac{2 \log{\left(\sin{\left(x + \frac{\pi}{2} \right)} \right)}}{x^{2}}\right) \sin^{\frac{2}{x}}{\left(x + \frac{\pi}{2} \right)} 
               /       /       2/x\\                  \
             2 |       |1 - tan |-||                  |
             - |       |        \2/|                  |
             x |  2*log|-----------|                  |
/       2/x\\  |       |       2/x\|            /x\   |
|1 - tan |-||  |       |1 + tan |-||       4*tan|-|   |
|        \2/|  |       \        \2//            \2/   |
|-----------| *|- ------------------ - ---------------|
|       2/x\|  |           2             /       2/x\\|
|1 + tan |-||  |          x            x*|1 - tan |-|||
\        \2//  \                         \        \2///
(1−tan⁡2(x2)tan⁡2(x2)+1)2x(−4tan⁡(x2)x(1−tan⁡2(x2))−2log⁡(1−tan⁡2(x2)tan⁡2(x2)+1)x2)\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{2}{x}} \left(- \frac{4 \tan{\left(\frac{x}{2} \right)}}{x \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)} - \frac{2 \log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x^{2}}\right) 
        2                             
        -                             
        x /  2*log(cos(x))   2*sin(x)\
(cos(x)) *|- ------------- - --------|
          |         2        x*cos(x)|
          \        x                 /
(−2sin⁡(x)xcos⁡(x)−2log⁡(cos⁡(x))x2)cos⁡2x(x)\left(- \frac{2 \sin{\left(x \right)}}{x \cos{\left(x \right)}} - \frac{2 \log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) \cos^{\frac{2}{x}}{\left(x \right)} 
           2 /                 /    pi\\
           - |              cos|x - --||
           x |log(cos(x))      \    2 /|
-2*(cos(x)) *|----------- + -----------|
             \     x           cos(x)  /
----------------------------------------
                   x
−2(cos⁡(x−π2)cos⁡(x)+log⁡(cos⁡(x))x)cos⁡2x(x)x- \frac{2 \left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} + \frac{\log{\left(\cos{\left(x \right)} \right)}}{x}\right) \cos^{\frac{2}{x}}{\left(x \right)}}{x} 
                  /   /       2/x\\         \
                2 |   |1 - tan |-||         |
                - |   |        \2/|         |
                x |log|-----------|         |
   /       2/x\\  |   |       2/x\|         |
   |1 - tan |-||  |   |1 + tan |-||         |
   |        \2/|  |   \        \2//         |
-2*|-----------| *|---------------- + tan(x)|
   |       2/x\|  \       x                 /
   |1 + tan |-||                             
   \        \2//                             
---------------------------------------------
                      x
−2(1−tan⁡2(x2)tan⁡2(x2)+1)2x(tan⁡(x)+log⁡(1−tan⁡2(x2)tan⁡2(x2)+1)x)x- \frac{2 \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{2}{x}} \left(\tan{\left(x \right)} + \frac{\log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x}\right)}{x} 
                /       /        2/x\\                   \
              2 |       |-1 + cot |-||                   |
              - |       |         \2/|                   |
              x |  2*log|------------|                   |
/        2/x\\  |       |       2/x\ |            /x\    |
|-1 + cot |-||  |       |1 + cot |-| |       4*cot|-|    |
|         \2/|  |       \        \2/ /            \2/    |
|------------| *|- ------------------- - ----------------|
|       2/x\ |  |            2             /        2/x\\|
|1 + cot |-| |  |           x            x*|-1 + cot |-|||
\        \2/ /  \                          \         \2///
(cot⁡2(x2)−1cot⁡2(x2)+1)2x(−4cot⁡(x2)x(cot⁡2(x2)−1)−2log⁡(cot⁡2(x2)−1cot⁡2(x2)+1)x2)\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{2}{x}} \left(- \frac{4 \cot{\left(\frac{x}{2} \right)}}{x \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)} - \frac{2 \log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x^{2}}\right) 
                2                               
                - /   /   /    pi\\            \
                x |log|sin|x + --||        2   |
   /   /    pi\\  |   \   \    2 //   2*sin (x)|
-2*|sin|x + --|| *|---------------- + ---------|
   \   \    2 //  \       x            sin(2*x)/
------------------------------------------------
                       x
−2(2sin⁡2(x)sin⁡(2x)+log⁡(sin⁡(x+π2))x)sin⁡2x(x+π2)x- \frac{2 \left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{\log{\left(\sin{\left(x + \frac{\pi}{2} \right)} \right)}}{x}\right) \sin^{\frac{2}{x}}{\left(x + \frac{\pi}{2} \right)}}{x} 
-2*sin(x + pi/2)^(2/x)*(log(sin(x + pi/2))/x + 2*sin(x)^2/sin(2*x))/x
Denominador racional [src] 
        2                                         
        -                                         
        x /     2                                \
(cos(x)) *\- 2*x *sin(x) - 2*x*cos(x)*log(cos(x))/
--------------------------------------------------
                     3                            
                    x *cos(x)
(−2x2sin⁡(x)−2xlog⁡(cos⁡(x))cos⁡(x))cos⁡2x(x)x3cos⁡(x)\frac{\left(- 2 x^{2} \sin{\left(x \right)} - 2 x \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{\frac{2}{x}}{\left(x \right)}}{x^{3} \cos{\left(x \right)}} 
cos(x)^(2/x)*(-2*x^2*sin(x) - 2*x*cos(x)*log(cos(x)))/(x^3*cos(x))
Compilar la expresión [src] 
        2                             
        -                             
        x /  2*log(cos(x))   2*sin(x)\
(cos(x)) *|- ------------- - --------|
          |         2        x*cos(x)|
          \        x                 /
(−2sin⁡(x)xcos⁡(x)−2log⁡(cos⁡(x))x2)cos⁡2x(x)\left(- \frac{2 \sin{\left(x \right)}}{x \cos{\left(x \right)}} - \frac{2 \log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) \cos^{\frac{2}{x}}{\left(x \right)} 
cos(x)^(2/x)*(-2*log(cos(x))/x^2 - 2*sin(x)/(x*cos(x)))
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