Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
log(2*x)^cos(x)*(-log(log(2*x))*sin(x)+cos(x)/(x*log(2*x)))
¿Cómo vas a descomponer esta log(2*x)^cos(x)*(-log(log(2*x))*sin(x)+cos(x)/(x*log(2*x))) expresión en fracciones?
Solución
Ha introducido
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cos(x) / cos(x) \
log (2*x)*|-log(log(2*x))*sin(x) + ----------|
\ x*log(2*x)/
$$\left(- \log{\left(\log{\left(2 x \right)} \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x \log{\left(2 x \right)}}\right) \log{\left(2 x \right)}^{\cos{\left(x \right)}}$$
log(2*x)^cos(x)*((-log(log(2*x)))*sin(x) + cos(x)/((x*log(2*x))))
Simplificación general
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-1 + cos(x)
log (2*x)*(-x*log(2*x)*log(log(2*x))*sin(x) + cos(x))
---------------------------------------------------------------
x
$$\frac{\left(- x \log{\left(2 x \right)} \log{\left(\log{\left(2 x \right)} \right)} \sin{\left(x \right)} + \cos{\left(x \right)}\right) \log{\left(2 x \right)}^{\cos{\left(x \right)} - 1}}{x}$$
log(2*x)^(-1 + cos(x))*(-x*log(2*x)*log(log(2*x))*sin(x) + cos(x))/x
Respuesta numérica
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log(2*x)^cos(x)*(-log(log(2*x))*sin(x) + cos(x)/(x*log(2*x)))
log(2*x)^cos(x)*(-log(log(2*x))*sin(x) + cos(x)/(x*log(2*x)))
Unión de expresiones racionales
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cos(x)
log (2*x)*(-x*log(2*x)*log(log(2*x))*sin(x) + cos(x))
----------------------------------------------------------
x*log(2*x)
$$\frac{\left(- x \log{\left(2 x \right)} \log{\left(\log{\left(2 x \right)} \right)} \sin{\left(x \right)} + \cos{\left(x \right)}\right) \log{\left(2 x \right)}^{\cos{\left(x \right)}}}{x \log{\left(2 x \right)}}$$
log(2*x)^cos(x)*(-x*log(2*x)*log(log(2*x))*sin(x) + cos(x))/(x*log(2*x))
Denominador racional
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cos(x)
log (2*x)*(-x*log(2*x)*log(log(2*x))*sin(x) + cos(x))
----------------------------------------------------------
x*log(2*x)
$$\frac{\left(- x \log{\left(2 x \right)} \log{\left(\log{\left(2 x \right)} \right)} \sin{\left(x \right)} + \cos{\left(x \right)}\right) \log{\left(2 x \right)}^{\cos{\left(x \right)}}}{x \log{\left(2 x \right)}}$$
log(2*x)^cos(x)*(-x*log(2*x)*log(log(2*x))*sin(x) + cos(x))/(x*log(2*x))
Denominador común
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cos(x)
(log(2) + log(x)) *cos(x) cos(x)
------------------------------ - (log(2) + log(x)) *log(log(2) + log(x))*sin(x)
x*log(2) + x*log(x)
$$- \left(\log{\left(x \right)} + \log{\left(2 \right)}\right)^{\cos{\left(x \right)}} \log{\left(\log{\left(x \right)} + \log{\left(2 \right)} \right)} \sin{\left(x \right)} + \frac{\left(\log{\left(x \right)} + \log{\left(2 \right)}\right)^{\cos{\left(x \right)}} \cos{\left(x \right)}}{x \log{\left(x \right)} + x \log{\left(2 \right)}}$$
(log(2) + log(x))^cos(x)*cos(x)/(x*log(2) + x*log(x)) - (log(2) + log(x))^cos(x)*log(log(2) + log(x))*sin(x)
Potencias
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I*x -I*x / I*x -I*x \
e e |e e |
---- + ----- |---- + ----- / -I*x I*x\ |
2 2 | 2 2 I*\- e + e /*log(log(2*x))|
(log(2*x)) *|------------ + --------------------------------|
\ x*log(2*x) 2 /
$$\left(\frac{i \left(e^{i x} - e^{- i x}\right) \log{\left(\log{\left(2 x \right)} \right)}}{2} + \frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{x \log{\left(2 x \right)}}\right) \log{\left(2 x \right)}^{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}$$
log(2*x)^(exp(i*x)/2 + exp(-i*x)/2)*((exp(i*x)/2 + exp(-i*x)/2)/(x*log(2*x)) + i*(-exp(-i*x) + exp(i*x))*log(log(2*x))/2)
Combinatoria
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cos(x)
-(log(2) + log(x)) *(-cos(x) + x*log(2)*log(log(2) + log(x))*sin(x) + x*log(x)*log(log(2) + log(x))*sin(x))
-----------------------------------------------------------------------------------------------------------------
x*(log(2) + log(x))
$$- \frac{\left(\log{\left(x \right)} + \log{\left(2 \right)}\right)^{\cos{\left(x \right)}} \left(x \log{\left(x \right)} \log{\left(\log{\left(x \right)} + \log{\left(2 \right)} \right)} \sin{\left(x \right)} + x \log{\left(2 \right)} \log{\left(\log{\left(x \right)} + \log{\left(2 \right)} \right)} \sin{\left(x \right)} - \cos{\left(x \right)}\right)}{x \left(\log{\left(x \right)} + \log{\left(2 \right)}\right)}$$
-(log(2) + log(x))^cos(x)*(-cos(x) + x*log(2)*log(log(2) + log(x))*sin(x) + x*log(x)*log(log(2) + log(x))*sin(x))/(x*(log(2) + log(x)))
Parte trigonométrica
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2/x\
1 - tan |-|
\2/
-----------
2/x\ / /x\ 2/x\ \
1 + tan |-| | 2*log(log(2*x))*tan|-| 1 - tan |-| |
\2/ | \2/ \2/ |
(log(2*x)) *|- ---------------------- + ------------------------|
| 2/x\ / 2/x\\ |
| 1 + tan |-| x*|1 + tan |-||*log(2*x)|
\ \2/ \ \2// /
$$\left(- \frac{2 \log{\left(\log{\left(2 x \right)} \right)} \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{x \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \log{\left(2 x \right)}}\right) \log{\left(2 x \right)}^{\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}}$$
1
------
sec(x) / log(log(2*x)) 1 \
(log(2*x)) *|- ------------- + -----------------|
\ csc(x) x*log(2*x)*sec(x)/
$$\left(- \frac{\log{\left(\log{\left(2 x \right)} \right)}}{\csc{\left(x \right)}} + \frac{1}{x \log{\left(2 x \right)} \sec{\left(x \right)}}\right) \log{\left(2 x \right)}^{\frac{1}{\sec{\left(x \right)}}}$$
/ pi\ / / pi\\
sin|x + --| | sin|x + --||
\ 2 / | \ 2 /|
(log(2*x)) *|-log(log(2*x))*sin(x) + -----------|
\ x*log(2*x)/
$$\left(- \log{\left(\log{\left(2 x \right)} \right)} \sin{\left(x \right)} + \frac{\sin{\left(x + \frac{\pi}{2} \right)}}{x \log{\left(2 x \right)}}\right) \log{\left(2 x \right)}^{\sin{\left(x + \frac{\pi}{2} \right)}}$$
cos(x) / / pi\ cos(x) \
log (2*x)*|- cos|x - --|*log(log(2*x)) + ----------|
\ \ 2 / x*log(2*x)/
$$\left(- \log{\left(\log{\left(2 x \right)} \right)} \cos{\left(x - \frac{\pi}{2} \right)} + \frac{\cos{\left(x \right)}}{x \log{\left(2 x \right)}}\right) \log{\left(2 x \right)}^{\cos{\left(x \right)}}$$
2/x\
-1 + cot |-|
\2/
------------
2/x\ / /x\ 2/x\ \
1 + cot |-| | 2*cot|-|*log(log(2*x)) -1 + cot |-| |
\2/ | \2/ \2/ |
(log(2*x)) *|- ---------------------- + ------------------------|
| 2/x\ / 2/x\\ |
| 1 + cot |-| x*|1 + cot |-||*log(2*x)|
\ \2/ \ \2// /
$$\left(- \frac{2 \log{\left(\log{\left(2 x \right)} \right)} \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{x \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \log{\left(2 x \right)}}\right) \log{\left(2 x \right)}^{\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}}$$
1
-----------
/pi \
csc|-- - x|
\2 / / log(log(2*x)) 1 \
(log(2*x)) *|- ------------- + ----------------------|
| csc(x) /pi \ |
| x*csc|-- - x|*log(2*x)|
\ \2 / /
$$\left(- \frac{\log{\left(\log{\left(2 x \right)} \right)}}{\csc{\left(x \right)}} + \frac{1}{x \log{\left(2 x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}\right) \log{\left(2 x \right)}^{\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}}$$
cos(x) / 1 \
log (2*x)*|----------*cos(x) - log(log(2*x))*sin(x)|
\x*log(2*x) /
$$\left(- \log{\left(\log{\left(2 x \right)} \right)} \sin{\left(x \right)} + \frac{1}{x \log{\left(2 x \right)}} \cos{\left(x \right)}\right) \log{\left(2 x \right)}^{\cos{\left(x \right)}}$$
1
------
sec(x) / log(log(2*x)) 1 \
(log(2*x)) *|- ------------- + -----------------|
| / pi\ x*log(2*x)*sec(x)|
| sec|x - --| |
\ \ 2 / /
$$\left(- \frac{\log{\left(\log{\left(2 x \right)} \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{x \log{\left(2 x \right)} \sec{\left(x \right)}}\right) \log{\left(2 x \right)}^{\frac{1}{\sec{\left(x \right)}}}$$
log(2*x)^(1/sec(x))*(-log(log(2*x))/sec(x - pi/2) + 1/(x*log(2*x)*sec(x)))