Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
log(cos(x)+1)/4-log(cos(x)-1)/4+cos(x)/(2*cos(x)^2-2)
¿Cómo vas a descomponer esta log(cos(x)+1)/4-log(cos(x)-1)/4+cos(x)/(2*cos(x)^2-2) expresión en fracciones?
Solución
Ha introducido
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log(cos(x) + 1) log(cos(x) - 1) cos(x)
--------------- - --------------- + -------------
4 4 2
2*cos (x) - 2
$$\left(- \frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{4} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{4}\right) + \frac{\cos{\left(x \right)}}{2 \cos^{2}{\left(x \right)} - 2}$$
log(cos(x) + 1)/4 - log(cos(x) - 1)/4 + cos(x)/(2*cos(x)^2 - 2)
Simplificación general
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log(-1 + cos(x)) log(1 + cos(x)) cos(x)
- ---------------- + --------------- - ---------
4 4 2
2*sin (x)
$$- \frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{4} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{4} - \frac{\cos{\left(x \right)}}{2 \sin^{2}{\left(x \right)}}$$
-log(-1 + cos(x))/4 + log(1 + cos(x))/4 - cos(x)/(2*sin(x)^2)
Potencias
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/ I*x -I*x\ / I*x -I*x\ I*x -I*x
| e e | | e e | e e
log|-1 + ---- + -----| log|1 + ---- + -----| ---- + -----
\ 2 2 / \ 2 2 / 2 2
- ---------------------- + --------------------- + ----------------------
4 4 2
/ I*x -I*x\
|e e |
-2 + 2*|---- + -----|
\ 2 2 /
$$- \frac{\log{\left(\frac{e^{i x}}{2} - 1 + \frac{e^{- i x}}{2} \right)}}{4} + \frac{\log{\left(\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2} \right)}}{4} + \frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{2 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} - 2}$$
log(-1 + cos(x)) log(1 + cos(x)) cos(x)
- ---------------- + --------------- + --------------
4 4 2
-2 + 2*cos (x)
$$- \frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{4} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{4} + \frac{\cos{\left(x \right)}}{2 \cos^{2}{\left(x \right)} - 2}$$
-log(-1 + cos(x))/4 + log(1 + cos(x))/4 + cos(x)/(-2 + 2*cos(x)^2)
Unión de expresiones racionales
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/ 2 \
2*cos(x) + \-1 + cos (x)/*(-log(-1 + cos(x)) + log(1 + cos(x)))
---------------------------------------------------------------
/ 2 \
4*\-1 + cos (x)/
$$\frac{\left(- \log{\left(\cos{\left(x \right)} - 1 \right)} + \log{\left(\cos{\left(x \right)} + 1 \right)}\right) \left(\cos^{2}{\left(x \right)} - 1\right) + 2 \cos{\left(x \right)}}{4 \left(\cos^{2}{\left(x \right)} - 1\right)}$$
(2*cos(x) + (-1 + cos(x)^2)*(-log(-1 + cos(x)) + log(1 + cos(x))))/(4*(-1 + cos(x)^2))
Combinatoria
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2 2
-log(1 + cos(x)) + 2*cos(x) + cos (x)*log(1 + cos(x)) - cos (x)*log(-1 + cos(x)) + log(-1 + cos(x))
---------------------------------------------------------------------------------------------------
4*(1 + cos(x))*(-1 + cos(x))
$$\frac{- \log{\left(\cos{\left(x \right)} - 1 \right)} \cos^{2}{\left(x \right)} + \log{\left(\cos{\left(x \right)} - 1 \right)} + \log{\left(\cos{\left(x \right)} + 1 \right)} \cos^{2}{\left(x \right)} - \log{\left(\cos{\left(x \right)} + 1 \right)} + 2 \cos{\left(x \right)}}{4 \left(\cos{\left(x \right)} - 1\right) \left(\cos{\left(x \right)} + 1\right)}$$
(-log(1 + cos(x)) + 2*cos(x) + cos(x)^2*log(1 + cos(x)) - cos(x)^2*log(-1 + cos(x)) + log(-1 + cos(x)))/(4*(1 + cos(x))*(-1 + cos(x)))
Denominador racional
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2 2
-log(1 + cos(x)) + 2*cos(x) + cos (x)*log(1 + cos(x)) - cos (x)*log(-1 + cos(x)) + log(-1 + cos(x))
---------------------------------------------------------------------------------------------------
2
-4 + 4*cos (x)
$$\frac{- \log{\left(\cos{\left(x \right)} - 1 \right)} \cos^{2}{\left(x \right)} + \log{\left(\cos{\left(x \right)} - 1 \right)} + \log{\left(\cos{\left(x \right)} + 1 \right)} \cos^{2}{\left(x \right)} - \log{\left(\cos{\left(x \right)} + 1 \right)} + 2 \cos{\left(x \right)}}{4 \cos^{2}{\left(x \right)} - 4}$$
(-log(1 + cos(x)) + 2*cos(x) + cos(x)^2*log(1 + cos(x)) - cos(x)^2*log(-1 + cos(x)) + log(-1 + cos(x)))/(-4 + 4*cos(x)^2)
Denominador común
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log(-1 + cos(x)) log(1 + cos(x)) cos(x)
- ---------------- + --------------- + --------------
4 4 2
-2 + 2*cos (x)
$$- \frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{4} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{4} + \frac{\cos{\left(x \right)}}{2 \cos^{2}{\left(x \right)} - 2}$$
-log(-1 + cos(x))/4 + log(1 + cos(x))/4 + cos(x)/(-2 + 2*cos(x)^2)
Compilar la expresión
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log(cos(x) - 1) log(cos(x) + 1) cos(x)
- --------------- + --------------- + --------------
4 4 2
-2 + 2*cos (x)
$$- \frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{4} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{4} + \frac{\cos{\left(x \right)}}{2 \cos^{2}{\left(x \right)} - 2}$$
-log(cos(x) - 1)/4 + log(cos(x) + 1)/4 + cos(x)/(-2 + 2*cos(x)^2)
Parte trigonométrica
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/ 1 \ / 1 \
log|-1 + ------| log|1 + ------|
\ sec(x)/ \ sec(x)/ 2/ pi\ / 1 1 \
- ---------------- + --------------- + sec |x - --|*|- - + ------------|
4 4 \ 2 / | 2 2/x pi\|
| sec |- - --||
\ \2 2 //
$$\left(- \frac{1}{2} + \frac{1}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}\right) \sec^{2}{\left(x - \frac{\pi}{2} \right)} - \frac{\log{\left(-1 + \frac{1}{\sec{\left(x \right)}} \right)}}{4} + \frac{\log{\left(1 + \frac{1}{\sec{\left(x \right)}} \right)}}{4}$$
log(-1 + cos(x)) log(1 + cos(x)) cos(x)
- ---------------- + --------------- - ---------
4 4 2
2*sin (x)
$$- \frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{4} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{4} - \frac{\cos{\left(x \right)}}{2 \sin^{2}{\left(x \right)}}$$
/ / pi\\ / / pi\\ / pi\
log|-1 + sin|x + --|| log|1 + sin|x + --|| sin|x + --|
\ \ 2 // \ \ 2 // \ 2 /
- --------------------- + -------------------- - -----------
4 4 2
2*sin (x)
$$- \frac{\log{\left(\sin{\left(x + \frac{\pi}{2} \right)} - 1 \right)}}{4} + \frac{\log{\left(\sin{\left(x + \frac{\pi}{2} \right)} + 1 \right)}}{4} - \frac{\sin{\left(x + \frac{\pi}{2} \right)}}{2 \sin^{2}{\left(x \right)}}$$
/ 2/x\ \
/ 2/x\\ / 2/x\\ 2 | 4*tan |-| |
| 1 - tan |-|| | 1 - tan |-|| / 2/x\\ | 1 \4/ |
| \2/| | \2/| |1 + tan |-|| *|- - + --------------|
log|-1 + -----------| log|1 + -----------| \ \2// | 2 2|
| 2/x\| | 2/x\| | / 2/x\\ |
| 1 + tan |-|| | 1 + tan |-|| | |1 + tan |-|| |
\ \2// \ \2// \ \ \4// /
- --------------------- + -------------------- + -------------------------------------
4 4 2/x\
4*tan |-|
\2/
$$\frac{\left(- \frac{1}{2} + \frac{4 \tan^{2}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{4 \tan^{2}{\left(\frac{x}{2} \right)}} - \frac{\log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - 1 \right)}}{4} + \frac{\log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1 \right)}}{4}$$
log(-1 + cos(x)) log(1 + cos(x)) cos(x)
- ---------------- + --------------- - --------------
4 4 2/ pi\
2*cos |x - --|
\ 2 /
$$- \frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{4} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{4} - \frac{\cos{\left(x \right)}}{2 \cos^{2}{\left(x - \frac{\pi}{2} \right)}}$$
/ 1 \ / 1 \
log|-1 + -----------| log|1 + -----------|
| /pi \| | /pi \|
| csc|-- - x|| | csc|-- - x||
\ \2 // \ \2 // 2 / 1 1 \
- --------------------- + -------------------- + csc (x)*|- - + -------|
4 4 | 2 2/x\|
| csc |-||
\ \2//
$$\left(- \frac{1}{2} + \frac{1}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc^{2}{\left(x \right)} - \frac{\log{\left(-1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{4} + \frac{\log{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{4}$$
/ / pi\\ / / pi\\ 1 2/x\
log|-1 + sin|x + --|| log|1 + sin|x + --|| - - + sin |-|
\ \ 2 // \ \ 2 // 2 \2/
- --------------------- + -------------------- + -------------
4 4 2
sin (x)
$$\frac{\sin^{2}{\left(\frac{x}{2} \right)} - \frac{1}{2}}{\sin^{2}{\left(x \right)}} - \frac{\log{\left(\sin{\left(x + \frac{\pi}{2} \right)} - 1 \right)}}{4} + \frac{\log{\left(\sin{\left(x + \frac{\pi}{2} \right)} + 1 \right)}}{4}$$
/ 1 \ / 1 \ 2/ pi\
log|-1 + ------| log|1 + ------| sec |x - --|
\ sec(x)/ \ sec(x)/ \ 2 /
- ---------------- + --------------- - ------------
4 4 2*sec(x)
$$- \frac{\log{\left(-1 + \frac{1}{\sec{\left(x \right)}} \right)}}{4} + \frac{\log{\left(1 + \frac{1}{\sec{\left(x \right)}} \right)}}{4} - \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{2 \sec{\left(x \right)}}$$
/ 2/x\ \
/ 2/x\\ / 2/x\\ 2 | 4*cot |-| |
| -1 + cot |-|| | -1 + cot |-|| / 2/x\\ | 1 \4/ |
| \2/| | \2/| |1 + cot |-|| *|- - + --------------|
log|-1 + ------------| log|1 + ------------| \ \2// | 2 2|
| 2/x\ | | 2/x\ | | / 2/x\\ |
| 1 + cot |-| | | 1 + cot |-| | | |1 + cot |-|| |
\ \2/ / \ \2/ / \ \ \4// /
- ---------------------- + --------------------- + -------------------------------------
4 4 2/x\
4*cot |-|
\2/
$$\frac{\left(- \frac{1}{2} + \frac{4 \cot^{2}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{4 \cot^{2}{\left(\frac{x}{2} \right)}} - \frac{\log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - 1 \right)}}{4} + \frac{\log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1 \right)}}{4}$$
/ 1 \ / 1 \
log|-1 + -----------| log|1 + -----------|
| /pi \| | /pi \|
| csc|-- - x|| | csc|-- - x|| 2
\ \2 // \ \2 // csc (x)
- --------------------- + -------------------- - -------------
4 4 /pi \
2*csc|-- - x|
\2 /
$$- \frac{\log{\left(-1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{4} + \frac{\log{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{4} - \frac{\csc^{2}{\left(x \right)}}{2 \csc{\left(- x + \frac{\pi}{2} \right)}}$$
/ 1 \ / 1 \
log|-1 + -----------| log|1 + -----------|
| /pi \| | /pi \|
| csc|-- - x|| | csc|-- - x||
\ \2 // \ \2 // 1
- --------------------- + -------------------- + -------------------------------
4 4 / 2 \ /pi \
|-2 + ------------|*csc|-- - x|
| 2/pi \| \2 /
| csc |-- - x||
\ \2 //
$$- \frac{\log{\left(-1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{4} + \frac{\log{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{4} + \frac{1}{\left(-2 + \frac{2}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}}$$
/ 2/x\\ / 2/x\\
| -1 + cot |-|| | -1 + cot |-||
| \2/| | \2/|
log|-1 + ------------| log|1 + ------------|
| 2/x\ | | 2/x\ | 2/x\
| 1 + cot |-| | | 1 + cot |-| | -1 + cot |-|
\ \2/ / \ \2/ / \2/
- ---------------------- + --------------------- + --------------------------------------
4 4 / 2\
| / 2/x\\ |
| 2*|-1 + cot |-|| |
/ 2/x\\ | \ \2// |
|1 + cot |-||*|-2 + -----------------|
\ \2// | 2 |
| / 2/x\\ |
| |1 + cot |-|| |
\ \ \2// /
$$- \frac{\log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - 1 \right)}}{4} + \frac{\log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1 \right)}}{4} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\left(\frac{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - 2\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ 1 \ / 1 \
log|-1 + ------| log|1 + ------|
\ sec(x)/ \ sec(x)/ 1
- ---------------- + --------------- + ---------------------
4 4 / 2 \
|-2 + -------|*sec(x)
| 2 |
\ sec (x)/
$$- \frac{\log{\left(-1 + \frac{1}{\sec{\left(x \right)}} \right)}}{4} + \frac{\log{\left(1 + \frac{1}{\sec{\left(x \right)}} \right)}}{4} + \frac{1}{\left(-2 + \frac{2}{\sec^{2}{\left(x \right)}}\right) \sec{\left(x \right)}}$$
1 2/x\
- - + sin |-|
log(-1 + cos(x)) log(1 + cos(x)) 2 \2/
- ---------------- + --------------- + -------------
4 4 2
sin (x)
$$\frac{\sin^{2}{\left(\frac{x}{2} \right)} - \frac{1}{2}}{\sin^{2}{\left(x \right)}} - \frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{4} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{4}$$
log(-1 + cos(x)) log(1 + cos(x)) cos(x)
- ---------------- + --------------- + --------------
4 4 2
-2 + 2*cos (x)
$$- \frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{4} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{4} + \frac{\cos{\left(x \right)}}{2 \cos^{2}{\left(x \right)} - 2}$$
/ 2/x\\ / 2/x\\
| 1 - tan |-|| | 1 - tan |-||
| \2/| | \2/|
log|-1 + -----------| log|1 + -----------|
| 2/x\| | 2/x\| 2/x\
| 1 + tan |-|| | 1 + tan |-|| 1 - tan |-|
\ \2// \ \2// \2/
- --------------------- + -------------------- + -------------------------------------
4 4 / 2\
| / 2/x\\ |
| 2*|1 - tan |-|| |
/ 2/x\\ | \ \2// |
|1 + tan |-||*|-2 + ----------------|
\ \2// | 2 |
| / 2/x\\ |
| |1 + tan |-|| |
\ \ \2// /
$$\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - 2\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{\log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - 1 \right)}}{4} + \frac{\log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1 \right)}}{4}$$
/ 2/x\\ / 2/x\\
| -1 + cot |-|| | -1 + cot |-||
| \2/| | \2/|
log|-1 + ------------| log|1 + ------------|
| 2/x\ | | 2/x\ | / 2/x\\ / 2/x\\
| 1 + cot |-| | | 1 + cot |-| | |1 + cot |-||*|-1 + cot |-||
\ \2/ / \ \2/ / \ \2// \ \2//
- ---------------------- + --------------------- - ----------------------------
4 4 2/x\
8*cot |-|
\2/
$$- \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}{8 \cot^{2}{\left(\frac{x}{2} \right)}} - \frac{\log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - 1 \right)}}{4} + \frac{\log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1 \right)}}{4}$$
1 2/x pi\
- - + cos |- - --|
log(-1 + cos(x)) log(1 + cos(x)) 2 \2 2 /
- ---------------- + --------------- + ------------------
4 4 2/ pi\
cos |x - --|
\ 2 /
$$\frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} - \frac{1}{2}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} - \frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{4} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{4}$$
/ 2/x\\ / 2/x\\
| 1 - tan |-|| | 1 - tan |-||
| \2/| | \2/|
log|-1 + -----------| log|1 + -----------|
| 2/x\| | 2/x\| / 2/x\\ / 2/x\\
| 1 + tan |-|| | 1 + tan |-|| |1 + tan |-||*|1 - tan |-||
\ \2// \ \2// \ \2// \ \2//
- --------------------- + -------------------- - ---------------------------
4 4 2/x\
8*tan |-|
\2/
$$- \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{8 \tan^{2}{\left(\frac{x}{2} \right)}} - \frac{\log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - 1 \right)}}{4} + \frac{\log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1 \right)}}{4}$$
/ / pi\\ / / pi\\ / pi\
log|-1 + sin|x + --|| log|1 + sin|x + --|| sin|x + --|
\ \ 2 // \ \ 2 // \ 2 /
- --------------------- + -------------------- + -------------------
4 4 2/ pi\
-2 + 2*sin |x + --|
\ 2 /
$$- \frac{\log{\left(\sin{\left(x + \frac{\pi}{2} \right)} - 1 \right)}}{4} + \frac{\log{\left(\sin{\left(x + \frac{\pi}{2} \right)} + 1 \right)}}{4} + \frac{\sin{\left(x + \frac{\pi}{2} \right)}}{2 \sin^{2}{\left(x + \frac{\pi}{2} \right)} - 2}$$
-log(-1 + sin(x + pi/2))/4 + log(1 + sin(x + pi/2))/4 + sin(x + pi/2)/(-2 + 2*sin(x + pi/2)^2)
Respuesta numérica
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0.25*log(cos(x) + 1) - 0.25*log(cos(x) - 1) + cos(x)/(-2.0 + 2.0*cos(x)^2)
0.25*log(cos(x) + 1) - 0.25*log(cos(x) - 1) + cos(x)/(-2.0 + 2.0*cos(x)^2)