Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(x)/(x-sin(x))+(-1+cos(x))*(2-cos(x))/(x-sin(x))^2
¿Cómo vas a descomponer esta sin(x)/(x-sin(x))+(-1+cos(x))*(2-cos(x))/(x-sin(x))^2 expresión en fracciones?
Solución
Ha introducido
[src]
sin(x) (-1 + cos(x))*(2 - cos(x))
---------- + --------------------------
x - sin(x) 2
(x - sin(x))
$$\frac{\left(2 - \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} - 1\right)}{\left(x - \sin{\left(x \right)}\right)^{2}} + \frac{\sin{\left(x \right)}}{x - \sin{\left(x \right)}}$$
sin(x)/(x - sin(x)) + ((-1 + cos(x))*(2 - cos(x)))/(x - sin(x))^2
Simplificación general
[src]
-3 + 3*cos(x) + x*sin(x)
------------------------
2
(x - sin(x))
$$\frac{x \sin{\left(x \right)} + 3 \cos{\left(x \right)} - 3}{\left(x - \sin{\left(x \right)}\right)^{2}}$$
(-3 + 3*cos(x) + x*sin(x))/(x - sin(x))^2
Respuesta numérica
[src]
sin(x)/(x - sin(x)) + (2.0 - cos(x))*(-1.0 + cos(x))/(x - sin(x))^2
sin(x)/(x - sin(x)) + (2.0 - cos(x))*(-1.0 + cos(x))/(x - sin(x))^2
Denominador racional
[src]
2
(x - sin(x)) *sin(x) + (-1 + cos(x))*(2 - cos(x))*(x - sin(x))
--------------------------------------------------------------
3
(x - sin(x))
$$\frac{\left(2 - \cos{\left(x \right)}\right) \left(x - \sin{\left(x \right)}\right) \left(\cos{\left(x \right)} - 1\right) + \left(x - \sin{\left(x \right)}\right)^{2} \sin{\left(x \right)}}{\left(x - \sin{\left(x \right)}\right)^{3}}$$
((x - sin(x))^2*sin(x) + (-1 + cos(x))*(2 - cos(x))*(x - sin(x)))/(x - sin(x))^3
Potencias
[src]
/ I*x -I*x\ / I*x -I*x\
| e e | | e e |
|-1 + ---- + -----|*|2 - ---- - -----| / -I*x I*x\
\ 2 2 / \ 2 2 / I*\- e + e /
-------------------------------------- - --------------------------
2 / / -I*x I*x\\
/ / -I*x I*x\\ | I*\- e + e /|
| I*\- e + e /| 2*|x + ------------------|
|x + ------------------| \ 2 /
\ 2 /
$$- \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(x + \frac{i \left(e^{i x} - e^{- i x}\right)}{2}\right)} + \frac{\left(- \frac{e^{i x}}{2} + 2 - \frac{e^{- i x}}{2}\right) \left(\frac{e^{i x}}{2} - 1 + \frac{e^{- i x}}{2}\right)}{\left(x + \frac{i \left(e^{i x} - e^{- i x}\right)}{2}\right)^{2}}$$
(-1 + exp(i*x)/2 + exp(-i*x)/2)*(2 - exp(i*x)/2 - exp(-i*x)/2)/(x + i*(-exp(-i*x) + exp(i*x))/2)^2 - i*(-exp(-i*x) + exp(i*x))/(2*(x + i*(-exp(-i*x) + exp(i*x))/2))
Combinatoria
[src]
2 2
-2 - cos (x) - sin (x) + 3*cos(x) + x*sin(x)
--------------------------------------------
2
(x - sin(x))
$$\frac{x \sin{\left(x \right)} - \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)} - 2}{\left(x - \sin{\left(x \right)}\right)^{2}}$$
(-2 - cos(x)^2 - sin(x)^2 + 3*cos(x) + x*sin(x))/(x - sin(x))^2
Unión de expresiones racionales
[src]
(-1 + cos(x))*(2 - cos(x)) + (x - sin(x))*sin(x)
------------------------------------------------
2
(x - sin(x))
$$\frac{\left(2 - \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} - 1\right) + \left(x - \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\left(x - \sin{\left(x \right)}\right)^{2}}$$
((-1 + cos(x))*(2 - cos(x)) + (x - sin(x))*sin(x))/(x - sin(x))^2
Parte trigonométrica
[src]
/ pi\
-3 + 3*sin|x + --| + x*sin(x)
\ 2 /
-----------------------------
2
(x - sin(x))
$$\frac{x \sin{\left(x \right)} + 3 \sin{\left(x + \frac{\pi}{2} \right)} - 3}{\left(x - \sin{\left(x \right)}\right)^{2}}$$
3 x
-3 + ------ + -----------
sec(x) / pi\
sec|x - --|
\ 2 /
-------------------------
2
/ 1 \
|x - -----------|
| / pi\|
| sec|x - --||
\ \ 2 //
$$\frac{\frac{x}{\sec{\left(x - \frac{\pi}{2} \right)}} - 3 + \frac{3}{\sec{\left(x \right)}}}{\left(x - \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2}}$$
/ pi\
cos|x - --|
\ 2 / (-1 + cos(x))*(2 - cos(x))
--------------- + --------------------------
/ pi\ 2
x - cos|x - --| / / pi\\
\ 2 / |x - cos|x - --||
\ \ 2 //
$$\frac{\left(2 - \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} - 1\right)}{\left(x - \cos{\left(x - \frac{\pi}{2} \right)}\right)^{2}} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{x - \cos{\left(x - \frac{\pi}{2} \right)}}$$
/ 2/x\\ /x\
3*|-1 + cot |-|| 2*x*cot|-|
\ \2// \2/
-3 + ---------------- + -----------
2/x\ 2/x\
1 + cot |-| 1 + cot |-|
\2/ \2/
-----------------------------------
2
/ /x\ \
| 2*cot|-| |
| \2/ |
|x - -----------|
| 2/x\|
| 1 + cot |-||
\ \2//
$$\frac{\frac{2 x \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{3 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - 3}{\left(x - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}}$$
3 x
-3 + ----------- + ------
/pi \ csc(x)
csc|-- - x|
\2 /
-------------------------
2
/ 1 \
|x - ------|
\ csc(x)/
$$\frac{\frac{x}{\csc{\left(x \right)}} - 3 + \frac{3}{\csc{\left(- x + \frac{\pi}{2} \right)}}}{\left(x - \frac{1}{\csc{\left(x \right)}}\right)^{2}}$$
/ 2/x\\ / 2/x\\
| -1 + cot |-|| | -1 + cot |-||
| \2/| | \2/|
|-1 + ------------|*|2 - ------------|
| 2/x\ | | 2/x\ | /x\
| 1 + cot |-| | | 1 + cot |-| | 2*cot|-|
\ \2/ / \ \2/ / \2/
-------------------------------------- + -------------------------------
2 / /x\ \
/ /x\ \ | 2*cot|-| |
| 2*cot|-| | / 2/x\\ | \2/ |
| \2/ | |1 + cot |-||*|x - -----------|
|x - -----------| \ \2// | 2/x\|
| 2/x\| | 1 + cot |-||
| 1 + cot |-|| \ \2//
\ \2//
$$\frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(x - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 2\right) \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - 1\right)}{\left(x - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}}$$
/ 1 \ / 1 \
|-1 + ------|*|2 - ------|
1 \ sec(x)/ \ sec(x)/
------------------- + --------------------------
/ 1 \ 2
|x - ------|*csc(x) / 1 \
\ csc(x)/ |x - ------|
\ csc(x)/
$$\frac{\left(-1 + \frac{1}{\sec{\left(x \right)}}\right) \left(2 - \frac{1}{\sec{\left(x \right)}}\right)}{\left(x - \frac{1}{\csc{\left(x \right)}}\right)^{2}} + \frac{1}{\left(x - \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}}$$
/ / pi\\ / / pi\\
|-1 + sin|x + --||*|2 - sin|x + --||
sin(x) \ \ 2 // \ \ 2 //
---------- + ------------------------------------
x - sin(x) 2
(x - sin(x))
$$\frac{\left(2 - \sin{\left(x + \frac{\pi}{2} \right)}\right) \left(\sin{\left(x + \frac{\pi}{2} \right)} - 1\right)}{\left(x - \sin{\left(x \right)}\right)^{2}} + \frac{\sin{\left(x \right)}}{x - \sin{\left(x \right)}}$$
/ 2/x\\ / 2/x\\
| 1 - tan |-|| | 1 - tan |-||
| \2/| | \2/|
|-1 + -----------|*|2 - -----------|
| 2/x\| | 2/x\| /x\
| 1 + tan |-|| | 1 + tan |-|| 2*tan|-|
\ \2// \ \2// \2/
------------------------------------ + -------------------------------
2 / /x\ \
/ /x\ \ | 2*tan|-| |
| 2*tan|-| | / 2/x\\ | \2/ |
| \2/ | |1 + tan |-||*|x - -----------|
|x - -----------| \ \2// | 2/x\|
| 2/x\| | 1 + tan |-||
| 1 + tan |-|| \ \2//
\ \2//
$$\frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(x - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{\left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 2\right) \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - 1\right)}{\left(x - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}}$$
/ pi\
-3 + 3*cos(x) + x*cos|x - --|
\ 2 /
-----------------------------
2
/ / pi\\
|x - cos|x - --||
\ \ 2 //
$$\frac{x \cos{\left(x - \frac{\pi}{2} \right)} + 3 \cos{\left(x \right)} - 3}{\left(x - \cos{\left(x - \frac{\pi}{2} \right)}\right)^{2}}$$
-3 + 3*cos(x) + x*sin(x)
------------------------
2
(x - sin(x))
$$\frac{x \sin{\left(x \right)} + 3 \cos{\left(x \right)} - 3}{\left(x - \sin{\left(x \right)}\right)^{2}}$$
/ 2/x\\ /x\
3*|1 - tan |-|| 2*x*tan|-|
\ \2// \2/
-3 + --------------- + -----------
2/x\ 2/x\
1 + tan |-| 1 + tan |-|
\2/ \2/
----------------------------------
2
/ /x\ \
| 2*tan|-| |
| \2/ |
|x - -----------|
| 2/x\|
| 1 + tan |-||
\ \2//
$$\frac{\frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{3 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - 3}{\left(x - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}}$$
/ 1 \ / 1 \
|-1 + ------|*|2 - ------|
1 \ sec(x)/ \ sec(x)/
----------------------------- + --------------------------
/ 1 \ / pi\ 2
|x - -----------|*sec|x - --| / 1 \
| / pi\| \ 2 / |x - -----------|
| sec|x - --|| | / pi\|
\ \ 2 // | sec|x - --||
\ \ 2 //
$$\frac{\left(-1 + \frac{1}{\sec{\left(x \right)}}\right) \left(2 - \frac{1}{\sec{\left(x \right)}}\right)}{\left(x - \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2}} + \frac{1}{\left(x - \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}}$$
/ 1 \ / 1 \
|-1 + -----------|*|2 - -----------|
| /pi \| | /pi \|
| csc|-- - x|| | csc|-- - x||
1 \ \2 // \ \2 //
------------------- + ------------------------------------
/ 1 \ 2
|x - ------|*csc(x) / 1 \
\ csc(x)/ |x - ------|
\ csc(x)/
$$\frac{\left(-1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \left(2 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)}{\left(x - \frac{1}{\csc{\left(x \right)}}\right)^{2}} + \frac{1}{\left(x - \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}}$$
1/((x - 1/csc(x))*csc(x)) + (-1 + 1/csc(pi/2 - x))*(2 - 1/csc(pi/2 - x))/(x - 1/csc(x))^2
Denominador común
[src]
2 2
-2 - cos (x) - sin (x) + 3*cos(x) + x*sin(x)
--------------------------------------------
2 2
x + sin (x) - 2*x*sin(x)
$$\frac{x \sin{\left(x \right)} - \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)} - 2}{x^{2} - 2 x \sin{\left(x \right)} + \sin^{2}{\left(x \right)}}$$
(-2 - cos(x)^2 - sin(x)^2 + 3*cos(x) + x*sin(x))/(x^2 + sin(x)^2 - 2*x*sin(x))