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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
x __________ /-log(cos(2*x))    2*sin(2*x)\
\/ cos(2*x) *|--------------- - ----------|
             |        2         x*cos(2*x)|
             \       x                    /
(−2sin⁡(2x)xcos⁡(2x)+(−1)log⁡(cos⁡(2x))x2)cos⁡1x(2x)\left(- \frac{2 \sin{\left(2 x \right)}}{x \cos{\left(2 x \right)}} + \frac{\left(-1\right) \log{\left(\cos{\left(2 x \right)} \right)}}{x^{2}}\right) \cos^{\frac{1}{x}}{\left(2 x \right)} 
cos(2*x)^(1/x)*((-log(cos(2*x)))/x^2 - 2*sin(2*x)/(x*cos(2*x)))
Simplificación general [src] 
                1                                         
           -1 + -                                         
                x                                         
-(cos(2*x))      *(cos(2*x)*log(cos(2*x)) + 2*x*sin(2*x)) 
----------------------------------------------------------
                             2                            
                            x
−(2xsin⁡(2x)+log⁡(cos⁡(2x))cos⁡(2x))cos⁡−1+1x(2x)x2- \frac{\left(2 x \sin{\left(2 x \right)} + \log{\left(\cos{\left(2 x \right)} \right)} \cos{\left(2 x \right)}\right) \cos^{-1 + \frac{1}{x}}{\left(2 x \right)}}{x^{2}} 
-cos(2*x)^(-1 + 1/x)*(cos(2*x)*log(cos(2*x)) + 2*x*sin(2*x))/x^2
Unión de expresiones racionales [src] 
x __________                                         
\/ cos(2*x) *(-cos(2*x)*log(cos(2*x)) - 2*x*sin(2*x))
-----------------------------------------------------
                      2                              
                     x *cos(2*x)
(−2xsin⁡(2x)−log⁡(cos⁡(2x))cos⁡(2x))cos⁡1x(2x)x2cos⁡(2x)\frac{\left(- 2 x \sin{\left(2 x \right)} - \log{\left(\cos{\left(2 x \right)} \right)} \cos{\left(2 x \right)}\right) \cos^{\frac{1}{x}}{\left(2 x \right)}}{x^{2} \cos{\left(2 x \right)}} 
cos(2*x)^(1/x)*(-cos(2*x)*log(cos(2*x)) - 2*x*sin(2*x))/(x^2*cos(2*x))
Respuesta numérica [src] 
cos(2*x)^(1/x)*(-log(cos(2*x))/x^2 - 2.0*sin(2*x)/(x*cos(2*x)))
cos(2*x)^(1/x)*(-log(cos(2*x))/x^2 - 2.0*sin(2*x)/(x*cos(2*x)))
Denominador común [src] 
 /x __________                              x __________         \ 
-\\/ cos(2*x) *cos(2*x)*log(cos(2*x)) + 2*x*\/ cos(2*x) *sin(2*x)/ 
-------------------------------------------------------------------
                             2                                     
                            x *cos(2*x)
−2xsin⁡(2x)cos⁡1x(2x)+log⁡(cos⁡(2x))cos⁡(2x)cos⁡1x(2x)x2cos⁡(2x)- \frac{2 x \sin{\left(2 x \right)} \cos^{\frac{1}{x}}{\left(2 x \right)} + \log{\left(\cos{\left(2 x \right)} \right)} \cos{\left(2 x \right)} \cos^{\frac{1}{x}}{\left(2 x \right)}}{x^{2} \cos{\left(2 x \right)}} 
-(cos(2*x)^(1/x)*cos(2*x)*log(cos(2*x)) + 2*x*cos(2*x)^(1/x)*sin(2*x))/(x^2*cos(2*x))
Combinatoria [src] 
 x __________                                         
-\/ cos(2*x) *(cos(2*x)*log(cos(2*x)) + 2*x*sin(2*x)) 
------------------------------------------------------
                      2                               
                     x *cos(2*x)
−(2xsin⁡(2x)+log⁡(cos⁡(2x))cos⁡(2x))cos⁡1x(2x)x2cos⁡(2x)- \frac{\left(2 x \sin{\left(2 x \right)} + \log{\left(\cos{\left(2 x \right)} \right)} \cos{\left(2 x \right)}\right) \cos^{\frac{1}{x}}{\left(2 x \right)}}{x^{2} \cos{\left(2 x \right)}} 
-cos(2*x)^(1/x)*(cos(2*x)*log(cos(2*x)) + 2*x*sin(2*x))/(x^2*cos(2*x))
Potencias [src] 
                        /     / -2*I*x    2*I*x\                         \
     __________________ |     |e         e     |                         |
    /  -2*I*x    2*I*x  |  log|------- + ------|     /   -2*I*x    2*I*x\|
   /  e         e       |     \   2        2   /   I*\- e       + e     /|
x /   ------- + ------ *|- --------------------- + ----------------------|
\/       2        2     |             2               / -2*I*x    2*I*x\ |
                        |            x                |e         e     | |
                        |                           x*|------- + ------| |
                        \                             \   2        2   / /
(i(e2ix−e−2ix)x(e2ix2+e−2ix2)−log⁡(e2ix2+e−2ix2)x2)(e2ix2+e−2ix2)1x\left(\frac{i \left(e^{2 i x} - e^{- 2 i x}\right)}{x \left(\frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2}\right)} - \frac{\log{\left(\frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2} \right)}}{x^{2}}\right) \left(\frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2}\right)^{\frac{1}{x}} 
x __________ /  log(cos(2*x))   2*sin(2*x)\
\/ cos(2*x) *|- ------------- - ----------|
             |         2        x*cos(2*x)|
             \        x                   /
(−2sin⁡(2x)xcos⁡(2x)−log⁡(cos⁡(2x))x2)cos⁡1x(2x)\left(- \frac{2 \sin{\left(2 x \right)}}{x \cos{\left(2 x \right)}} - \frac{\log{\left(\cos{\left(2 x \right)} \right)}}{x^{2}}\right) \cos^{\frac{1}{x}}{\left(2 x \right)} 
cos(2*x)^(1/x)*(-log(cos(2*x))/x^2 - 2*sin(2*x)/(x*cos(2*x)))
Compilar la expresión [src] 
x __________ /  log(cos(2*x))   2*sin(2*x)\
\/ cos(2*x) *|- ------------- - ----------|
             |         2        x*cos(2*x)|
             \        x                   /
(−2sin⁡(2x)xcos⁡(2x)−log⁡(cos⁡(2x))x2)cos⁡1x(2x)\left(- \frac{2 \sin{\left(2 x \right)}}{x \cos{\left(2 x \right)}} - \frac{\log{\left(\cos{\left(2 x \right)} \right)}}{x^{2}}\right) \cos^{\frac{1}{x}}{\left(2 x \right)} 
cos(2*x)^(1/x)*(-log(cos(2*x))/x^2 - 2*sin(2*x)/(x*cos(2*x)))
Abrimos la expresión [src] 
     ________________                            ________________              
  x /           2        /          2   \     x /           2                  
  \/  -1 + 2*cos (x) *log\-1 + 2*cos (x)/   4*\/  -1 + 2*cos (x) *cos(x)*sin(x)
- --------------------------------------- - -----------------------------------
                      2                                           2            
                     x                                -x + 2*x*cos (x)
−4(2cos⁡2(x)−1)1xsin⁡(x)cos⁡(x)2xcos⁡2(x)−x−(2cos⁡2(x)−1)1xlog⁡(2cos⁡2(x)−1)x2- \frac{4 \left(2 \cos^{2}{\left(x \right)} - 1\right)^{\frac{1}{x}} \sin{\left(x \right)} \cos{\left(x \right)}}{2 x \cos^{2}{\left(x \right)} - x} - \frac{\left(2 \cos^{2}{\left(x \right)} - 1\right)^{\frac{1}{x}} \log{\left(2 \cos^{2}{\left(x \right)} - 1 \right)}}{x^{2}} 
x __________ /-log(cos(2*x))    2*sin(2*x)\
\/ cos(2*x) *|--------------- - ----------|
             |        2         x*cos(2*x)|
             \       x                    /
((−1)log⁡(cos⁡(2x))x2−2sin⁡(2x)xcos⁡(2x))cos⁡1x(2x)\left(\frac{\left(-1\right) \log{\left(\cos{\left(2 x \right)} \right)}}{x^{2}} - \frac{2 \sin{\left(2 x \right)}}{x \cos{\left(2 x \right)}}\right) \cos^{\frac{1}{x}}{\left(2 x \right)} 
cos(2*x)^(1/x)*((-log(cos(2*x)))/x^2 - 2*sin(2*x)/(x*cos(2*x)))
Denominador racional [src] 
x __________ /     2                                    \
\/ cos(2*x) *\- 2*x *sin(2*x) - x*cos(2*x)*log(cos(2*x))/
---------------------------------------------------------
                        3                                
                       x *cos(2*x)
(−2x2sin⁡(2x)−xlog⁡(cos⁡(2x))cos⁡(2x))cos⁡1x(2x)x3cos⁡(2x)\frac{\left(- 2 x^{2} \sin{\left(2 x \right)} - x \log{\left(\cos{\left(2 x \right)} \right)} \cos{\left(2 x \right)}\right) \cos^{\frac{1}{x}}{\left(2 x \right)}}{x^{3} \cos{\left(2 x \right)}} 
cos(2*x)^(1/x)*(-2*x^2*sin(2*x) - x*cos(2*x)*log(cos(2*x)))/(x^3*cos(2*x))
Parte trigonométrica [src] 
                     /                /       2   \\ 
                     |                |1 - tan (x)|| 
       _____________ |             log|-----------|| 
      /        2     |                |       2   || 
     /  1 - tan (x)  |                \1 + tan (x)/| 
-   /   ----------- *|2*tan(2*x) + ----------------| 
 x /           2     \                    x        / 
 \/     1 + tan (x)                                  
-----------------------------------------------------
                          x
−(1−tan⁡2(x)tan⁡2(x)+1)1x(2tan⁡(2x)+log⁡(1−tan⁡2(x)tan⁡2(x)+1)x)x- \frac{\left(\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}\right)^{\frac{1}{x}} \left(2 \tan{\left(2 x \right)} + \frac{\log{\left(\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} \right)}}{x}\right)}{x} 
x __________ /  log(cos(2*x))   2*sin(2*x)\
\/ cos(2*x) *|- ------------- - ----------|
             |         2        x*cos(2*x)|
             \        x                   /
(−2sin⁡(2x)xcos⁡(2x)−log⁡(cos⁡(2x))x2)cos⁡1x(2x)\left(- \frac{2 \sin{\left(2 x \right)}}{x \cos{\left(2 x \right)}} - \frac{\log{\left(\cos{\left(2 x \right)} \right)}}{x^{2}}\right) \cos^{\frac{1}{x}}{\left(2 x \right)} 
             /                       /      pi\\
             |                  2*cos|2*x - --||
x __________ |  log(cos(2*x))        \      2 /|
\/ cos(2*x) *|- ------------- - ---------------|
             |         2           x*cos(2*x)  |
             \        x                        /
(−2cos⁡(2x−π2)xcos⁡(2x)−log⁡(cos⁡(2x))x2)cos⁡1x(2x)\left(- \frac{2 \cos{\left(2 x - \frac{\pi}{2} \right)}}{x \cos{\left(2 x \right)}} - \frac{\log{\left(\cos{\left(2 x \right)} \right)}}{x^{2}}\right) \cos^{\frac{1}{x}}{\left(2 x \right)} 
               /     /   1    \                  \
    __________ |  log|--------|                  |
   /    1      |     \sec(2*x)/      2*sec(2*x)  |
x /  -------- *|- ------------- - ---------------|
\/   sec(2*x)  |         2             /      pi\|
               |        x         x*sec|2*x - --||
               \                       \      2 //
(−2sec⁡(2x)xsec⁡(2x−π2)−log⁡(1sec⁡(2x))x2)(1sec⁡(2x))1x\left(- \frac{2 \sec{\left(2 x \right)}}{x \sec{\left(2 x - \frac{\pi}{2} \right)}} - \frac{\log{\left(\frac{1}{\sec{\left(2 x \right)}} \right)}}{x^{2}}\right) \left(\frac{1}{\sec{\left(2 x \right)}}\right)^{\frac{1}{x}} 
                       /   /      1      \                  \ 
                       |log|-------------|                  | 
                       |   |   /pi      \|        /pi      \| 
       _______________ |   |csc|-- - 2*x||   2*csc|-- - 2*x|| 
      /       1        |   \   \2       //        \2       /| 
-    /  ------------- *|------------------ + ---------------| 
    /      /pi      \  \        x                csc(2*x)   / 
 x /    csc|-- - 2*x|                                         
 \/        \2       /                                         
--------------------------------------------------------------
                              x
−(2csc⁡(−2x+π2)csc⁡(2x)+log⁡(1csc⁡(−2x+π2))x)(1csc⁡(−2x+π2))1xx- \frac{\left(\frac{2 \csc{\left(- 2 x + \frac{\pi}{2} \right)}}{\csc{\left(2 x \right)}} + \frac{\log{\left(\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} \right)}}{x}\right) \left(\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}\right)^{\frac{1}{x}}}{x} 
                      /     /      1      \                  \
                      |  log|-------------|                  |
                      |     |   /pi      \|        /pi      \|
      _______________ |     |csc|-- - 2*x||   2*csc|-- - 2*x||
     /       1        |     \   \2       //        \2       /|
    /  ------------- *|- ------------------ - ---------------|
   /      /pi      \  |           2              x*csc(2*x)  |
x /    csc|-- - 2*x|  \          x                           /
\/        \2       /
(−2csc⁡(−2x+π2)xcsc⁡(2x)−log⁡(1csc⁡(−2x+π2))x2)(1csc⁡(−2x+π2))1x\left(- \frac{2 \csc{\left(- 2 x + \frac{\pi}{2} \right)}}{x \csc{\left(2 x \right)}} - \frac{\log{\left(\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} \right)}}{x^{2}}\right) \left(\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}\right)^{\frac{1}{x}} 
                    /     /       2   \                  \
                    |     |1 - tan (x)|                  |
      _____________ |  log|-----------|                  |
     /        2     |     |       2   |                  |
    /  1 - tan (x)  |     \1 + tan (x)/       4*tan(x)   |
   /   ----------- *|- ---------------- - ---------------|
x /           2     |          2            /       2   \|
\/     1 + tan (x)  \         x           x*\1 - tan (x)//
(1−tan⁡2(x)tan⁡2(x)+1)1x(−4tan⁡(x)x(1−tan⁡2(x))−log⁡(1−tan⁡2(x)tan⁡2(x)+1)x2)\left(\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}\right)^{\frac{1}{x}} \left(- \frac{4 \tan{\left(x \right)}}{x \left(1 - \tan^{2}{\left(x \right)}\right)} - \frac{\log{\left(\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} \right)}}{x^{2}}\right) 
                     /   /   /pi      \\              \ 
     _______________ |log|sin|-- + 2*x||        2     | 
    /    /pi      \  |   \   \2       //   4*sin (2*x)| 
-x /  sin|-- + 2*x| *|------------------ + -----------| 
 \/      \2       /  \        x              sin(4*x) / 
--------------------------------------------------------
                           x
−(4sin⁡2(2x)sin⁡(4x)+log⁡(sin⁡(2x+π2))x)sin⁡1x(2x+π2)x- \frac{\left(\frac{4 \sin^{2}{\left(2 x \right)}}{\sin{\left(4 x \right)}} + \frac{\log{\left(\sin{\left(2 x + \frac{\pi}{2} \right)} \right)}}{x}\right) \sin^{\frac{1}{x}}{\left(2 x + \frac{\pi}{2} \right)}}{x} 
                     /     /        2   \                   \
                     |     |-1 + cot (x)|                   |
      ______________ |  log|------------|                   |
     /         2     |     |       2    |                   |
    /  -1 + cot (x)  |     \1 + cot (x) /       4*cot(x)    |
   /   ------------ *|- ----------------- - ----------------|
x /           2      |           2            /        2   \|
\/     1 + cot (x)   \          x           x*\-1 + cot (x)//
(cot⁡2(x)−1cot⁡2(x)+1)1x(−4cot⁡(x)x(cot⁡2(x)−1)−log⁡(cot⁡2(x)−1cot⁡2(x)+1)x2)\left(\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1}\right)^{\frac{1}{x}} \left(- \frac{4 \cot{\left(x \right)}}{x \left(\cot^{2}{\left(x \right)} - 1\right)} - \frac{\log{\left(\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} \right)}}{x^{2}}\right) 
 x __________ /             log(cos(2*x))\ 
-\/ cos(2*x) *|2*tan(2*x) + -------------| 
              \                   x      / 
-------------------------------------------
                     x
−(2tan⁡(2x)+log⁡(cos⁡(2x))x)cos⁡1x(2x)x- \frac{\left(2 \tan{\left(2 x \right)} + \frac{\log{\left(\cos{\left(2 x \right)} \right)}}{x}\right) \cos^{\frac{1}{x}}{\left(2 x \right)}}{x} 
              /                     /      pi\\ 
              |                2*cos|2*x - --|| 
 x __________ |log(cos(2*x))        \      2 /| 
-\/ cos(2*x) *|------------- + ---------------| 
              \      x             cos(2*x)   / 
------------------------------------------------
                       x
−(2cos⁡(2x−π2)cos⁡(2x)+log⁡(cos⁡(2x))x)cos⁡1x(2x)x- \frac{\left(\frac{2 \cos{\left(2 x - \frac{\pi}{2} \right)}}{\cos{\left(2 x \right)}} + \frac{\log{\left(\cos{\left(2 x \right)} \right)}}{x}\right) \cos^{\frac{1}{x}}{\left(2 x \right)}}{x} 
                      /              /        2   \\ 
                      |              |-1 + cot (x)|| 
       ______________ |           log|------------|| 
      /         2     |              |       2    || 
     /  -1 + cot (x)  |   2          \1 + cot (x) /| 
-   /   ------------ *|-------- + -----------------| 
 x /           2      \cot(2*x)           x        / 
 \/     1 + cot (x)                                  
-----------------------------------------------------
                          x
−(cot⁡2(x)−1cot⁡2(x)+1)1x(2cot⁡(2x)+log⁡(cot⁡2(x)−1cot⁡2(x)+1)x)x- \frac{\left(\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1}\right)^{\frac{1}{x}} \left(\frac{2}{\cot{\left(2 x \right)}} + \frac{\log{\left(\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} \right)}}{x}\right)}{x} 
                /   /   1    \                \ 
     __________ |log|--------|                | 
    /    1      |   \sec(2*x)/     2*sec(2*x) | 
-x /  -------- *|------------- + -------------| 
 \/   sec(2*x)  |      x            /      pi\| 
                |                sec|2*x - --|| 
                \                   \      2 // 
------------------------------------------------
                       x
−(2sec⁡(2x)sec⁡(2x−π2)+log⁡(1sec⁡(2x))x)(1sec⁡(2x))1xx- \frac{\left(\frac{2 \sec{\left(2 x \right)}}{\sec{\left(2 x - \frac{\pi}{2} \right)}} + \frac{\log{\left(\frac{1}{\sec{\left(2 x \right)}} \right)}}{x}\right) \left(\frac{1}{\sec{\left(2 x \right)}}\right)^{\frac{1}{x}}}{x} 
                    /     /   /pi      \\                  \
    _______________ |  log|sin|-- + 2*x||                  |
   /    /pi      \  |     \   \2       //      2*sin(2*x)  |
x /  sin|-- + 2*x| *|- ------------------ - ---------------|
\/      \2       /  |           2                /pi      \|
                    |          x            x*sin|-- + 2*x||
                    \                            \2       //
(−2sin⁡(2x)xsin⁡(2x+π2)−log⁡(sin⁡(2x+π2))x2)sin⁡1x(2x+π2)\left(- \frac{2 \sin{\left(2 x \right)}}{x \sin{\left(2 x + \frac{\pi}{2} \right)}} - \frac{\log{\left(\sin{\left(2 x + \frac{\pi}{2} \right)} \right)}}{x^{2}}\right) \sin^{\frac{1}{x}}{\left(2 x + \frac{\pi}{2} \right)} 
               /     /   1    \             \
    __________ |  log|--------|             |
   /    1      |     \sec(2*x)/   2*sec(2*x)|
x /  -------- *|- ------------- - ----------|
\/   sec(2*x)  |         2        x*csc(2*x)|
               \        x                   /
(−2sec⁡(2x)xcsc⁡(2x)−log⁡(1sec⁡(2x))x2)(1sec⁡(2x))1x\left(- \frac{2 \sec{\left(2 x \right)}}{x \csc{\left(2 x \right)}} - \frac{\log{\left(\frac{1}{\sec{\left(2 x \right)}} \right)}}{x^{2}}\right) \left(\frac{1}{\sec{\left(2 x \right)}}\right)^{\frac{1}{x}} 
(1/sec(2*x))^(1/x)*(-log(1/sec(2*x))/x^2 - 2*sec(2*x)/(x*csc(2*x)))
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