Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
(1+cos(x))^(sqrt(x))*(log(1+cos(x))/(2*sqrt(x))-sqrt(x)*sin(x)/(1+cos(x)))
¿Cómo vas a descomponer esta (1+cos(x))^(sqrt(x))*(log(1+cos(x))/(2*sqrt(x))-sqrt(x)*sin(x)/(1+cos(x))) expresión en fracciones?
Solución
Ha introducido
[src]
___ / ___ \
\/ x |log(1 + cos(x)) \/ x *sin(x)|
(1 + cos(x)) *|--------------- - ------------|
| ___ 1 + cos(x) |
\ 2*\/ x /
$$\left(- \frac{\sqrt{x} \sin{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{2 \sqrt{x}}\right) \left(\cos{\left(x \right)} + 1\right)^{\sqrt{x}}$$
(1 + cos(x))^(sqrt(x))*(log(1 + cos(x))/((2*sqrt(x))) - sqrt(x)*sin(x)/(1 + cos(x)))
Simplificación general
[src]
___
-1 + \/ x /(1 + cos(x))*log(1 + cos(x)) \
(1 + cos(x)) *|---------------------------- - x*sin(x)|
\ 2 /
----------------------------------------------------------------
___
\/ x
$$\frac{\left(- x \sin{\left(x \right)} + \frac{\left(\cos{\left(x \right)} + 1\right) \log{\left(\cos{\left(x \right)} + 1 \right)}}{2}\right) \left(\cos{\left(x \right)} + 1\right)^{\sqrt{x} - 1}}{\sqrt{x}}$$
(1 + cos(x))^(-1 + sqrt(x))*((1 + cos(x))*log(1 + cos(x))/2 - x*sin(x))/sqrt(x)
Respuesta numérica
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(1.0 + cos(x))^(x^0.5)*(0.5*x^(-0.5)*log(1 + cos(x)) - x^0.5*sin(x)/(1.0 + cos(x)))
(1.0 + cos(x))^(x^0.5)*(0.5*x^(-0.5)*log(1 + cos(x)) - x^0.5*sin(x)/(1.0 + cos(x)))
Denominador común
[src]
/ ___ ___ ___ \
| \/ x \/ x \/ x |
-\- (1 + cos(x)) *log(1 + cos(x)) - (1 + cos(x)) *cos(x)*log(1 + cos(x)) + 2*x*(1 + cos(x)) *sin(x)/
-----------------------------------------------------------------------------------------------------------------
___ ___
2*\/ x + 2*\/ x *cos(x)
$$- \frac{2 x \left(\cos{\left(x \right)} + 1\right)^{\sqrt{x}} \sin{\left(x \right)} - \left(\cos{\left(x \right)} + 1\right)^{\sqrt{x}} \log{\left(\cos{\left(x \right)} + 1 \right)} \cos{\left(x \right)} - \left(\cos{\left(x \right)} + 1\right)^{\sqrt{x}} \log{\left(\cos{\left(x \right)} + 1 \right)}}{2 \sqrt{x} \cos{\left(x \right)} + 2 \sqrt{x}}$$
-(-(1 + cos(x))^(sqrt(x))*log(1 + cos(x)) - (1 + cos(x))^(sqrt(x))*cos(x)*log(1 + cos(x)) + 2*x*(1 + cos(x))^(sqrt(x))*sin(x))/(2*sqrt(x) + 2*sqrt(x)*cos(x))
Unión de expresiones racionales
[src]
___
\/ x
(1 + cos(x)) *((1 + cos(x))*log(1 + cos(x)) - 2*x*sin(x))
-------------------------------------------------------------
___
2*\/ x *(1 + cos(x))
$$\frac{\left(- 2 x \sin{\left(x \right)} + \left(\cos{\left(x \right)} + 1\right) \log{\left(\cos{\left(x \right)} + 1 \right)}\right) \left(\cos{\left(x \right)} + 1\right)^{\sqrt{x}}}{2 \sqrt{x} \left(\cos{\left(x \right)} + 1\right)}$$
(1 + cos(x))^(sqrt(x))*((1 + cos(x))*log(1 + cos(x)) - 2*x*sin(x))/(2*sqrt(x)*(1 + cos(x)))
Potencias
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___ / / I*x -I*x\ \
\/ x | | e e | |
/ I*x -I*x\ |log|1 + ---- + -----| ___ / -I*x I*x\|
| e e | | \ 2 2 / I*\/ x *\- e + e /|
|1 + ---- + -----| *|--------------------- + ------------------------|
\ 2 2 / | ___ / I*x -I*x\ |
| 2*\/ x | e e | |
| 2*|1 + ---- + -----| |
\ \ 2 2 / /
$$\left(\frac{i \sqrt{x} \left(e^{i x} - e^{- i x}\right)}{2 \left(\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}\right)} + \frac{\log{\left(\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2} \right)}}{2 \sqrt{x}}\right) \left(\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}\right)^{\sqrt{x}}$$
(1 + exp(i*x)/2 + exp(-i*x)/2)^(sqrt(x))*(log(1 + exp(i*x)/2 + exp(-i*x)/2)/(2*sqrt(x)) + i*sqrt(x)*(-exp(-i*x) + exp(i*x))/(2*(1 + exp(i*x)/2 + exp(-i*x)/2)))
Denominador racional
[src]
___ ___ ___
-1 + \/ x -1 + \/ x -1 + \/ x
(1 + cos(x)) *log(1 + cos(x)) + (1 + cos(x)) *cos(x)*log(1 + cos(x)) - 2*x*(1 + cos(x)) *sin(x)
--------------------------------------------------------------------------------------------------------------------------
___
2*\/ x
$$\frac{- 2 x \left(\cos{\left(x \right)} + 1\right)^{\sqrt{x} - 1} \sin{\left(x \right)} + \left(\cos{\left(x \right)} + 1\right)^{\sqrt{x} - 1} \log{\left(\cos{\left(x \right)} + 1 \right)} \cos{\left(x \right)} + \left(\cos{\left(x \right)} + 1\right)^{\sqrt{x} - 1} \log{\left(\cos{\left(x \right)} + 1 \right)}}{2 \sqrt{x}}$$
((1 + cos(x))^(-1 + sqrt(x))*log(1 + cos(x)) + (1 + cos(x))^(-1 + sqrt(x))*cos(x)*log(1 + cos(x)) - 2*x*(1 + cos(x))^(-1 + sqrt(x))*sin(x))/(2*sqrt(x))
Combinatoria
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___
\/ x
-(1 + cos(x)) *(-log(1 + cos(x)) - cos(x)*log(1 + cos(x)) + 2*x*sin(x))
----------------------------------------------------------------------------
___
2*\/ x *(1 + cos(x))
$$- \frac{\left(\cos{\left(x \right)} + 1\right)^{\sqrt{x}} \left(2 x \sin{\left(x \right)} - \log{\left(\cos{\left(x \right)} + 1 \right)} \cos{\left(x \right)} - \log{\left(\cos{\left(x \right)} + 1 \right)}\right)}{2 \sqrt{x} \left(\cos{\left(x \right)} + 1\right)}$$
-(1 + cos(x))^(sqrt(x))*(-log(1 + cos(x)) - cos(x)*log(1 + cos(x)) + 2*x*sin(x))/(2*sqrt(x)*(1 + cos(x)))
Parte trigonométrica
[src]
/ ___ / pi\\
___ | \/ x *cos|x - --||
\/ x |log(1 + cos(x)) \ 2 /|
(1 + cos(x)) *|--------------- - -----------------|
| ___ 1 + cos(x) |
\ 2*\/ x /
$$\left(- \frac{\sqrt{x} \cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} + 1} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{2 \sqrt{x}}\right) \left(\cos{\left(x \right)} + 1\right)^{\sqrt{x}}$$
___ / / 1 \ \
\/ x |log|1 + ------| ___ |
/ 1 \ | \ sec(x)/ \/ x |
|1 + ------| *|--------------- - -------------------|
\ sec(x)/ | ___ / 1 \ |
| 2*\/ x |1 + ------|*csc(x)|
\ \ sec(x)/ /
$$\left(1 + \frac{1}{\sec{\left(x \right)}}\right)^{\sqrt{x}} \left(- \frac{\sqrt{x}}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \csc{\left(x \right)}} + \frac{\log{\left(1 + \frac{1}{\sec{\left(x \right)}} \right)}}{2 \sqrt{x}}\right)$$
/ / 2/x\\ \
| | 1 - tan |-|| |
___ | | \2/| |
\/ x |log|1 + -----------| |
/ 2/x\\ | | 2/x\| ___ /x\ |
| 1 - tan |-|| | | 1 + tan |-|| 2*\/ x *tan|-| |
| \2/| | \ \2// \2/ |
|1 + -----------| *|-------------------- - -------------------------------|
| 2/x\| | ___ / 2/x\\|
| 1 + tan |-|| | 2*\/ x | 1 - tan |-|||
\ \2// | / 2/x\\ | \2/||
| |1 + tan |-||*|1 + -----------||
| \ \2// | 2/x\||
| | 1 + tan |-|||
\ \ \2///
$$\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)^{\sqrt{x}} \left(- \frac{2 \sqrt{x} \tan{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{\log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1 \right)}}{2 \sqrt{x}}\right)$$
/ / 1 \ \
|log|1 + -----------| |
___ | | /pi \| |
\/ x | | csc|-- - x|| ___ |
/ 1 \ | \ \2 // \/ x |
|1 + -----------| *|-------------------- - ------------------------|
| /pi \| | ___ / 1 \ |
| csc|-- - x|| | 2*\/ x |1 + -----------|*csc(x)|
\ \2 // | | /pi \| |
| | csc|-- - x|| |
\ \ \2 // /
$$\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{\sqrt{x}} \left(- \frac{\sqrt{x}}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(x \right)}} + \frac{\log{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{2 \sqrt{x}}\right)$$
___ / ___ \
\/ x | 1 \/ x *sin(x)|
(1 + cos(x)) *|-------*log(1 + cos(x)) - ------------|
| ___ 1 + cos(x) |
\2*\/ x /
$$\left(\frac{1}{2 \sqrt{x}} \log{\left(\cos{\left(x \right)} + 1 \right)} - \frac{\sqrt{x} \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right) \left(\cos{\left(x \right)} + 1\right)^{\sqrt{x}}$$
___ / / / pi\\ \
\/ x |log|1 + sin|x + --|| ___ |
/ / pi\\ | \ \ 2 // \/ x *sin(x) |
|1 + sin|x + --|| *|-------------------- - ---------------|
\ \ 2 // | ___ / pi\|
| 2*\/ x 1 + sin|x + --||
\ \ 2 //
$$\left(- \frac{\sqrt{x} \sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{2} \right)} + 1} + \frac{\log{\left(\sin{\left(x + \frac{\pi}{2} \right)} + 1 \right)}}{2 \sqrt{x}}\right) \left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right)^{\sqrt{x}}$$
/ / 2/x\\ \
| | -1 + cot |-|| |
___ | | \2/| |
\/ x |log|1 + ------------| |
/ 2/x\\ | | 2/x\ | ___ /x\ |
| -1 + cot |-|| | | 1 + cot |-| | 2*\/ x *cot|-| |
| \2/| | \ \2/ / \2/ |
|1 + ------------| *|--------------------- - --------------------------------|
| 2/x\ | | ___ / 2/x\\|
| 1 + cot |-| | | 2*\/ x | -1 + cot |-|||
\ \2/ / | / 2/x\\ | \2/||
| |1 + cot |-||*|1 + ------------||
| \ \2// | 2/x\ ||
| | 1 + cot |-| ||
\ \ \2/ //
$$\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)^{\sqrt{x}} \left(- \frac{2 \sqrt{x} \cot{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{\log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1 \right)}}{2 \sqrt{x}}\right)$$
___ / / 1 \ \
\/ x |log|1 + ------| ___ |
/ 1 \ | \ sec(x)/ \/ x |
|1 + ------| *|--------------- - ------------------------|
\ sec(x)/ | ___ / 1 \ / pi\|
| 2*\/ x |1 + ------|*sec|x - --||
\ \ sec(x)/ \ 2 //
$$\left(1 + \frac{1}{\sec{\left(x \right)}}\right)^{\sqrt{x}} \left(- \frac{\sqrt{x}}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{\log{\left(1 + \frac{1}{\sec{\left(x \right)}} \right)}}{2 \sqrt{x}}\right)$$
(1 + 1/sec(x))^(sqrt(x))*(log(1 + 1/sec(x))/(2*sqrt(x)) - sqrt(x)/((1 + 1/sec(x))*sec(x - pi/2)))