Sr Examen
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?
Solución
Ha introducido
[src]
2
-
x /-2*log(cos(x)) 2*sin(x)\
(cos(x)) *|-------------- - --------|
| 2 x*cos(x)|
\ 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
$$- \frac{2 \left(x \sin{\left(x \right)} + \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{-1 + \frac{2}{x}}{\left(x \right)}}{x^{2}}$$
-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 /
$$\left(- \frac{2 \sin{\left(x \right)}}{x \cos{\left(x \right)}} - \frac{2 \log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) \left(\cos^{2}{\left(x \right)}\right)^{\frac{1}{x}}$$
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 / /
$$\left(\frac{i \left(e^{i x} - e^{- i x}\right)}{x \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)} - \frac{2 \log{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2} \right)}}{x^{2}}\right) \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{\frac{2}{x}}$$
2
-
x / 2*log(cos(x)) 2*sin(x)\
(cos(x)) *|- ------------- - --------|
| 2 x*cos(x)|
\ x /
$$\left(- \frac{2 \sin{\left(x \right)}}{x \cos{\left(x \right)}} - \frac{2 \log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) \cos^{\frac{2}{x}}{\left(x \right)}$$
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)
$$- \frac{2 x \sin{\left(x \right)} \cos^{\frac{2}{x}}{\left(x \right)} + 2 \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} \cos^{\frac{2}{x}}{\left(x \right)}}{x^{2} \cos{\left(x \right)}}$$
-(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)
$$- \frac{2 \left(x \sin{\left(x \right)} + \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{\frac{2}{x}}{\left(x \right)}}{x^{2} \cos{\left(x \right)}}$$
-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)
$$\frac{2 \left(- x \sin{\left(x \right)} - \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{\frac{2}{x}}{\left(x \right)}}{x^{2} \cos{\left(x \right)}}$$
2*cos(x)^(2/x)*(-x*sin(x) - cos(x)*log(cos(x)))/(x^2*cos(x))
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
$$- \frac{2 \left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} + \frac{\log{\left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{x}\right) \left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{\frac{2}{x}}}{x}$$
2
-
x /log(cos(x)) \
-2*(cos(x)) *|----------- + tan(x)|
\ x /
-----------------------------------
x
$$- \frac{2 \left(\tan{\left(x \right)} + \frac{\log{\left(\cos{\left(x \right)} \right)}}{x}\right) \cos^{\frac{2}{x}}{\left(x \right)}}{x}$$
2
- / / 1 \ \
x | 2*log|------| |
/ 1 \ | \sec(x)/ 2*sec(x) |
|------| *|- ------------- - -------------|
\sec(x)/ | 2 / pi\|
| x x*sec|x - --||
\ \ 2 //
$$\left(- \frac{2 \sec{\left(x \right)}}{x \sec{\left(x - \frac{\pi}{2} \right)}} - \frac{2 \log{\left(\frac{1}{\sec{\left(x \right)}} \right)}}{x^{2}}\right) \left(\frac{1}{\sec{\left(x \right)}}\right)^{\frac{2}{x}}$$
2 / / pi\\
- | 2*cos|x - --||
x | 2*log(cos(x)) \ 2 /|
(cos(x)) *|- ------------- - -------------|
| 2 x*cos(x) |
\ x /
$$\left(- \frac{2 \cos{\left(x - \frac{\pi}{2} \right)}}{x \cos{\left(x \right)}} - \frac{2 \log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) \cos^{\frac{2}{x}}{\left(x \right)}$$
/ / 1 \ \
2 | 2*log|-----------| |
- | | /pi \| /pi \|
x | |csc|-- - x|| 2*csc|-- - x||
/ 1 \ | \ \2 // \2 /|
|-----------| *|- ------------------ - -------------|
| /pi \| | 2 x*csc(x) |
|csc|-- - x|| \ x /
\ \2 //
$$\left(- \frac{2 \csc{\left(- x + \frac{\pi}{2} \right)}}{x \csc{\left(x \right)}} - \frac{2 \log{\left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{x^{2}}\right) \left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{\frac{2}{x}}$$
2
- / / 1 \ \
x | 2*log|------| |
/ 1 \ | \sec(x)/ 2*sec(x)|
|------| *|- ------------- - --------|
\sec(x)/ | 2 x*csc(x)|
\ x /
$$\left(- \frac{2 \sec{\left(x \right)}}{x \csc{\left(x \right)}} - \frac{2 \log{\left(\frac{1}{\sec{\left(x \right)}} \right)}}{x^{2}}\right) \left(\frac{1}{\sec{\left(x \right)}}\right)^{\frac{2}{x}}$$
/ / 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
$$- \frac{2 \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{2}{x}} \left(\frac{1}{\cot{\left(x \right)}} + \frac{\log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x}\right)}{x}$$
2
- / / 1 \ \
x |log|------| |
/ 1 \ | \sec(x)/ sec(x) |
-2*|------| *|----------- + -----------|
\sec(x)/ | x / pi\|
| sec|x - --||
\ \ 2 //
----------------------------------------
x
$$- \frac{2 \left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{\log{\left(\frac{1}{\sec{\left(x \right)}} \right)}}{x}\right) \left(\frac{1}{\sec{\left(x \right)}}\right)^{\frac{2}{x}}}{x}$$
2
- / / / pi\\ \
x | 2*log|sin|x + --|| |
/ / pi\\ | \ \ 2 // 2*sin(x) |
|sin|x + --|| *|- ------------------ - -------------|
\ \ 2 // | 2 / pi\|
| x x*sin|x + --||
\ \ 2 //
$$\left(- \frac{2 \sin{\left(x \right)}}{x \sin{\left(x + \frac{\pi}{2} \right)}} - \frac{2 \log{\left(\sin{\left(x + \frac{\pi}{2} \right)} \right)}}{x^{2}}\right) \sin^{\frac{2}{x}}{\left(x + \frac{\pi}{2} \right)}$$
/ / 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///
$$\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{2}{x}} \left(- \frac{4 \tan{\left(\frac{x}{2} \right)}}{x \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)} - \frac{2 \log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x^{2}}\right)$$
2
-
x / 2*log(cos(x)) 2*sin(x)\
(cos(x)) *|- ------------- - --------|
| 2 x*cos(x)|
\ x /
$$\left(- \frac{2 \sin{\left(x \right)}}{x \cos{\left(x \right)}} - \frac{2 \log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) \cos^{\frac{2}{x}}{\left(x \right)}$$
2 / / pi\\
- | cos|x - --||
x |log(cos(x)) \ 2 /|
-2*(cos(x)) *|----------- + -----------|
\ x cos(x) /
----------------------------------------
x
$$- \frac{2 \left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} + \frac{\log{\left(\cos{\left(x \right)} \right)}}{x}\right) \cos^{\frac{2}{x}}{\left(x \right)}}{x}$$
/ / 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
$$- \frac{2 \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{2}{x}} \left(\tan{\left(x \right)} + \frac{\log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x}\right)}{x}$$
/ / 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///
$$\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{2}{x}} \left(- \frac{4 \cot{\left(\frac{x}{2} \right)}}{x \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)} - \frac{2 \log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x^{2}}\right)$$
2
- / / / pi\\ \
x |log|sin|x + --|| 2 |
/ / pi\\ | \ \ 2 // 2*sin (x)|
-2*|sin|x + --|| *|---------------- + ---------|
\ \ 2 // \ x sin(2*x)/
------------------------------------------------
x
$$- \frac{2 \left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{\log{\left(\sin{\left(x + \frac{\pi}{2} \right)} \right)}}{x}\right) \sin^{\frac{2}{x}}{\left(x + \frac{\pi}{2} \right)}}{x}$$
-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)
$$\frac{\left(- 2 x^{2} \sin{\left(x \right)} - 2 x \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{\frac{2}{x}}{\left(x \right)}}{x^{3} \cos{\left(x \right)}}$$
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 /
$$\left(- \frac{2 \sin{\left(x \right)}}{x \cos{\left(x \right)}} - \frac{2 \log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) \cos^{\frac{2}{x}}{\left(x \right)}$$
cos(x)^(2/x)*(-2*log(cos(x))/x^2 - 2*sin(x)/(x*cos(x)))