Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(x)/((1-cos(x))*(1+cos(x)))+log(1-cos(x))*sin(x)/(1+cos(x))^2
¿Cómo vas a descomponer esta sin(x)/((1-cos(x))*(1+cos(x)))+log(1-cos(x))*sin(x)/(1+cos(x))^2 expresión en fracciones?
Solución
Ha introducido
[src]
sin(x) log(1 - cos(x))*sin(x)
------------------------- + ----------------------
(1 - cos(x))*(1 + cos(x)) 2
(1 + cos(x))
$$\frac{\log{\left(1 - \cos{\left(x \right)} \right)} \sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} + \frac{\sin{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} + 1\right)}$$
sin(x)/(((1 - cos(x))*(1 + cos(x)))) + (log(1 - cos(x))*sin(x))/(1 + cos(x))^2
Simplificación general
[src]
1 - cos(x)*log(1 - cos(x)) + cos(x) + log(1 - cos(x))
-----------------------------------------------------
(1 + cos(x))*sin(x)
$$\frac{- \log{\left(1 - \cos{\left(x \right)} \right)} \cos{\left(x \right)} + \log{\left(1 - \cos{\left(x \right)} \right)} + \cos{\left(x \right)} + 1}{\left(\cos{\left(x \right)} + 1\right) \sin{\left(x \right)}}$$
(1 - cos(x)*log(1 - cos(x)) + cos(x) + log(1 - cos(x)))/((1 + cos(x))*sin(x))
Respuesta numérica
[src]
sin(x)/((1.0 - cos(x))*(1.0 + cos(x))) + log(1 - cos(x))*sin(x)/(1.0 + cos(x))^2
sin(x)/((1.0 - cos(x))*(1.0 + cos(x))) + log(1 - cos(x))*sin(x)/(1.0 + cos(x))^2
Unión de expresiones racionales
[src]
(1 + (1 - cos(x))*log(1 - cos(x)) + cos(x))*sin(x)
--------------------------------------------------
2
(1 - cos(x))*(1 + cos(x))
$$\frac{\left(\left(1 - \cos{\left(x \right)}\right) \log{\left(1 - \cos{\left(x \right)} \right)} + \cos{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} + 1\right)^{2}}$$
(1 + (1 - cos(x))*log(1 - cos(x)) + cos(x))*sin(x)/((1 - cos(x))*(1 + cos(x))^2)
Potencias
[src]
/ I*x -I*x\
/ -I*x I*x\ | e e |
/ -I*x I*x\ I*\- e + e /*log|1 - ---- - -----|
I*\- e + e / \ 2 2 /
- --------------------------------------- - ----------------------------------------
/ I*x -I*x\ / I*x -I*x\ 2
| e e | | e e | / I*x -I*x\
2*|1 + ---- + -----|*|1 - ---- - -----| | e e |
\ 2 2 / \ 2 2 / 2*|1 + ---- + -----|
\ 2 2 /
$$- \frac{i \left(e^{i x} - e^{- i x}\right) \log{\left(- \frac{e^{i x}}{2} + 1 - \frac{e^{- i x}}{2} \right)}}{2 \left(\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}\right)^{2}} - \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(- \frac{e^{i x}}{2} + 1 - \frac{e^{- i x}}{2}\right) \left(\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}\right)}$$
-i*(-exp(-i*x) + exp(i*x))/(2*(1 + exp(i*x)/2 + exp(-i*x)/2)*(1 - exp(i*x)/2 - exp(-i*x)/2)) - i*(-exp(-i*x) + exp(i*x))*log(1 - exp(i*x)/2 - exp(-i*x)/2)/(2*(1 + exp(i*x)/2 + exp(-i*x)/2)^2)
Denominador racional
[src]
2
(1 + cos(x)) *sin(x) + (1 - cos(x))*(1 + cos(x))*log(1 - cos(x))*sin(x)
-----------------------------------------------------------------------
3
(1 - cos(x))*(1 + cos(x))
$$\frac{\left(1 - \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} + 1\right) \log{\left(1 - \cos{\left(x \right)} \right)} \sin{\left(x \right)} + \left(\cos{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} + 1\right)^{3}}$$
((1 + cos(x))^2*sin(x) + (1 - cos(x))*(1 + cos(x))*log(1 - cos(x))*sin(x))/((1 - cos(x))*(1 + cos(x))^3)
Combinatoria
[src]
(-1 - cos(x) - log(1 - cos(x)) + cos(x)*log(1 - cos(x)))*sin(x)
---------------------------------------------------------------
2
(1 + cos(x)) *(-1 + cos(x))
$$\frac{\left(\log{\left(1 - \cos{\left(x \right)} \right)} \cos{\left(x \right)} - \log{\left(1 - \cos{\left(x \right)} \right)} - \cos{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right) \left(\cos{\left(x \right)} + 1\right)^{2}}$$
(-1 - cos(x) - log(1 - cos(x)) + cos(x)*log(1 - cos(x)))*sin(x)/((1 + cos(x))^2*(-1 + cos(x)))
Parte trigonométrica
[src]
log(1 - cos(x))*sin(x)
---------------------- + csc(x)
2
(1 + cos(x))
$$\csc{\left(x \right)} + \frac{\log{\left(1 - \cos{\left(x \right)} \right)} \sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}}$$
/ 2/x\\
| -1 + cot |-||
/x\ | \2/|
2*cot|-|*log|1 - ------------|
2/x\ \2/ | 2/x\ |
1 + cot |-| | 1 + cot |-| |
\2/ \ \2/ /
----------- + ---------------------------------
/x\ 2
2*cot|-| / 2/x\\
\2/ | -1 + cot |-||
/ 2/x\\ | \2/|
|1 + cot |-||*|1 + ------------|
\ \2// | 2/x\ |
| 1 + cot |-| |
\ \2/ /
$$\frac{\cot^{2}{\left(\frac{x}{2} \right)} + 1}{2 \cot{\left(\frac{x}{2} \right)}} + \frac{2 \log{\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1 \right)} \cot{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ 1 \
log|1 - -----------|
| /pi \|
| csc|-- - x||
\ \2 //
------------------------- + csc(x)
2
/ 1 \
|1 + -----------| *csc(x)
| /pi \|
| csc|-- - x||
\ \2 //
$$\csc{\left(x \right)} + \frac{\log{\left(1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2} \csc{\left(x \right)}}$$
1 log(1 - cos(x))*sin(x)
------ + ----------------------
sin(x) 2
(1 + cos(x))
$$\frac{1}{\sin{\left(x \right)}} + \frac{\log{\left(1 - \cos{\left(x \right)} \right)} \sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}}$$
/ 1 \
log|1 - ------|
1 \ sec(x)/
-------------------------------- + --------------------
/ 1 \ / 1 \ 2
|1 + ------|*|1 - ------|*csc(x) / 1 \
\ sec(x)/ \ sec(x)/ |1 + ------| *csc(x)
\ sec(x)/
$$\frac{\log{\left(1 - \frac{1}{\sec{\left(x \right)}} \right)}}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right)^{2} \csc{\left(x \right)}} + \frac{1}{\left(1 - \frac{1}{\sec{\left(x \right)}}\right) \left(1 + \frac{1}{\sec{\left(x \right)}}\right) \csc{\left(x \right)}}$$
/ 1 \
log|1 - ------|
1 \ sec(x)/
------------------------------------- + -------------------------
/ 1 \ / 1 \ / pi\ 2
|1 + ------|*|1 - ------|*sec|x - --| / 1 \ / pi\
\ sec(x)/ \ sec(x)/ \ 2 / |1 + ------| *sec|x - --|
\ sec(x)/ \ 2 /
$$\frac{\log{\left(1 - \frac{1}{\sec{\left(x \right)}} \right)}}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right)^{2} \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\left(1 - \frac{1}{\sec{\left(x \right)}}\right) \left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}}$$
/ pi\ / pi\
cos|x - --| cos|x - --|*log(1 - cos(x))
\ 2 / \ 2 /
------------------------- + ---------------------------
(1 - cos(x))*(1 + cos(x)) 2
(1 + cos(x))
$$\frac{\log{\left(1 - \cos{\left(x \right)} \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\left(1 - \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} + 1\right)}$$
/ 2/x\\
| 1 - tan |-||
| \2/| /x\
2*log|1 - -----------|*tan|-|
2/x\ | 2/x\| \2/
1 + tan |-| | 1 + tan |-||
\2/ \ \2//
----------- + --------------------------------
/x\ 2
2*tan|-| / 2/x\\
\2/ | 1 - tan |-||
/ 2/x\\ | \2/|
|1 + tan |-||*|1 + -----------|
\ \2// | 2/x\|
| 1 + tan |-||
\ \2//
$$\frac{\tan^{2}{\left(\frac{x}{2} \right)} + 1}{2 \tan{\left(\frac{x}{2} \right)}} + \frac{2 \log{\left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1 \right)} \tan{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ 2/x\\
| -1 + cot |-||
/x\ | \2/|
2*cot|-|*log|1 - ------------|
/x\ \2/ | 2/x\ |
2*cot|-| | 1 + cot |-| |
\2/ \ \2/ /
--------------------------------------------------- + ---------------------------------
/ 2/x\\ / 2/x\\ 2
| -1 + cot |-|| | -1 + cot |-|| / 2/x\\
/ 2/x\\ | \2/| | \2/| | -1 + cot |-||
|1 + cot |-||*|1 + ------------|*|1 - ------------| / 2/x\\ | \2/|
\ \2// | 2/x\ | | 2/x\ | |1 + cot |-||*|1 + ------------|
| 1 + cot |-| | | 1 + cot |-| | \ \2// | 2/x\ |
\ \2/ / \ \2/ / | 1 + cot |-| |
\ \2/ /
$$\frac{2 \log{\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1 \right)} \cot{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ pi\
cos|x - --|*log(1 - cos(x))
1 \ 2 /
----------- + ---------------------------
/ pi\ 2
cos|x - --| (1 + cos(x))
\ 2 /
$$\frac{1}{\cos{\left(x - \frac{\pi}{2} \right)}} + \frac{\log{\left(1 - \cos{\left(x \right)} \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}}$$
/ / pi\\
log|1 - sin|x + --||*sin(x)
sin(x) \ \ 2 //
----------------------------------- + ---------------------------
/ / pi\\ / / pi\\ 2
|1 - sin|x + --||*|1 + sin|x + --|| / / pi\\
\ \ 2 // \ \ 2 // |1 + sin|x + --||
\ \ 2 //
$$\frac{\log{\left(1 - \sin{\left(x + \frac{\pi}{2} \right)} \right)} \sin{\left(x \right)}}{\left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right)^{2}} + \frac{\sin{\left(x \right)}}{\left(1 - \sin{\left(x + \frac{\pi}{2} \right)}\right) \left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right)}$$
/ 1 \
log|1 - -----------|
| /pi \|
| csc|-- - x||
1 \ \2 //
------------------------------------------ + -------------------------
/ 1 \ / 1 \ 2
|1 + -----------|*|1 - -----------|*csc(x) / 1 \
| /pi \| | /pi \| |1 + -----------| *csc(x)
| csc|-- - x|| | csc|-- - x|| | /pi \|
\ \2 // \ \2 // | csc|-- - x||
\ \2 //
$$\frac{\log{\left(1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2} \csc{\left(x \right)}} + \frac{1}{\left(1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(x \right)}}$$
/ 2/x\\
| 1 - tan |-||
| \2/| /x\
2*log|1 - -----------|*tan|-|
/x\ | 2/x\| \2/
2*tan|-| | 1 + tan |-||
\2/ \ \2//
------------------------------------------------- + --------------------------------
/ 2/x\\ / 2/x\\ 2
| 1 - tan |-|| | 1 - tan |-|| / 2/x\\
/ 2/x\\ | \2/| | \2/| | 1 - tan |-||
|1 + tan |-||*|1 + -----------|*|1 - -----------| / 2/x\\ | \2/|
\ \2// | 2/x\| | 2/x\| |1 + tan |-||*|1 + -----------|
| 1 + tan |-|| | 1 + tan |-|| \ \2// | 2/x\|
\ \2// \ \2// | 1 + tan |-||
\ \2//
$$\frac{2 \log{\left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1 \right)} \tan{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ / pi\\
log|1 - sin|x + --||*sin(x)
1 \ \ 2 //
------ + ---------------------------
sin(x) 2
/ / pi\\
|1 + sin|x + --||
\ \ 2 //
$$\frac{1}{\sin{\left(x \right)}} + \frac{\log{\left(1 - \sin{\left(x + \frac{\pi}{2} \right)} \right)} \sin{\left(x \right)}}{\left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right)^{2}}$$
/ 1 \
log|1 - ------|
\ sec(x)/ / pi\
------------------------- + sec|x - --|
2 \ 2 /
/ 1 \ / pi\
|1 + ------| *sec|x - --|
\ sec(x)/ \ 2 /
$$\sec{\left(x - \frac{\pi}{2} \right)} + \frac{\log{\left(1 - \frac{1}{\sec{\left(x \right)}} \right)}}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right)^{2} \sec{\left(x - \frac{\pi}{2} \right)}}$$
log(1 - 1/sec(x))/((1 + 1/sec(x))^2*sec(x - pi/2)) + sec(x - pi/2)
Denominador común
[src]
-sin(x) - cos(x)*sin(x) - log(1 - cos(x))*sin(x) + cos(x)*log(1 - cos(x))*sin(x)
--------------------------------------------------------------------------------
2 3
-1 + cos (x) + cos (x) - cos(x)
$$\frac{\log{\left(1 - \cos{\left(x \right)} \right)} \sin{\left(x \right)} \cos{\left(x \right)} - \log{\left(1 - \cos{\left(x \right)} \right)} \sin{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)} - \sin{\left(x \right)}}{\cos^{3}{\left(x \right)} + \cos^{2}{\left(x \right)} - \cos{\left(x \right)} - 1}$$
(-sin(x) - cos(x)*sin(x) - log(1 - cos(x))*sin(x) + cos(x)*log(1 - cos(x))*sin(x))/(-1 + cos(x)^2 + cos(x)^3 - cos(x))