Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(x)/(1+sin(x))-(1-cos(x))*cos(x)/(1+sin(x))^2
¿Cómo vas a descomponer esta sin(x)/(1+sin(x))-(1-cos(x))*cos(x)/(1+sin(x))^2 expresión en fracciones?
Solución
Ha introducido
[src]
sin(x) (1 - cos(x))*cos(x)
---------- - -------------------
1 + sin(x) 2
(1 + sin(x))
$$- \frac{\left(1 - \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} + \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + 1}$$
sin(x)/(1 + sin(x)) - (1 - cos(x))*cos(x)/(1 + sin(x))^2
Simplificación general
[src]
___ / pi\
1 - \/ 2 *cos|x + --|
\ 4 /
---------------------
2
(1 + sin(x))
$$\frac{- \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} + 1}{\left(\sin{\left(x \right)} + 1\right)^{2}}$$
(1 - sqrt(2)*cos(x + pi/4))/(1 + sin(x))^2
Respuesta numérica
[src]
sin(x)/(1.0 + sin(x)) - (1.0 - cos(x))*cos(x)/(1.0 + sin(x))^2
sin(x)/(1.0 + sin(x)) - (1.0 - cos(x))*cos(x)/(1.0 + sin(x))^2
Denominador racional
[src]
2
(1 + sin(x)) *sin(x) - (1 - cos(x))*(1 + sin(x))*cos(x)
-------------------------------------------------------
3
(1 + sin(x))
$$\frac{- \left(1 - \cos{\left(x \right)}\right) \left(\sin{\left(x \right)} + 1\right) \cos{\left(x \right)} + \left(\sin{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{3}}$$
((1 + sin(x))^2*sin(x) - (1 - cos(x))*(1 + sin(x))*cos(x))/(1 + sin(x))^3
Denominador común
[src]
2
-1 + cos (x) - cos(x) - sin(x)
1 + ------------------------------
2
1 + sin (x) + 2*sin(x)
$$1 + \frac{- \sin{\left(x \right)} + \cos^{2}{\left(x \right)} - \cos{\left(x \right)} - 1}{\sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} + 1}$$
1 + (-1 + cos(x)^2 - cos(x) - sin(x))/(1 + sin(x)^2 + 2*sin(x))
Combinatoria
[src]
2 2
cos (x) + sin (x) - cos(x) + sin(x)
-----------------------------------
2
(1 + sin(x))
$$\frac{\sin^{2}{\left(x \right)} + \sin{\left(x \right)} + \cos^{2}{\left(x \right)} - \cos{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}}$$
(cos(x)^2 + sin(x)^2 - cos(x) + sin(x))/(1 + sin(x))^2
Unión de expresiones racionales
[src]
(1 + sin(x))*sin(x) - (1 - cos(x))*cos(x)
-----------------------------------------
2
(1 + sin(x))
$$\frac{- \left(1 - \cos{\left(x \right)}\right) \cos{\left(x \right)} + \left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}}$$
((1 + sin(x))*sin(x) - (1 - cos(x))*cos(x))/(1 + sin(x))^2
Potencias
[src]
/ I*x -I*x\ / I*x -I*x\
|e e | | e e |
|---- + -----|*|1 - ---- - -----| / -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*|1 - ------------------|
|1 - ------------------| \ 2 /
\ 2 /
$$- \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 1\right)} - \frac{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \left(- \frac{e^{i x}}{2} + 1 - \frac{e^{- i x}}{2}\right)}{\left(- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 1\right)^{2}}$$
-(exp(i*x)/2 + exp(-i*x)/2)*(1 - exp(i*x)/2 - exp(-i*x)/2)/(1 - i*(-exp(-i*x) + exp(i*x))/2)^2 - i*(-exp(-i*x) + exp(i*x))/(2*(1 - i*(-exp(-i*x) + exp(i*x))/2))
Parte trigonométrica
[src]
___ / 3*pi\
1 - \/ 2 *sin|x + ----|
\ 4 /
-----------------------
2
(1 + sin(x))
$$\frac{- \sqrt{2} \sin{\left(x + \frac{3 \pi}{4} \right)} + 1}{\left(\sin{\left(x \right)} + 1\right)^{2}}$$
___
\/ 2
1 - -----------
/ pi\
sec|x + --|
\ 4 /
------------------
2
/ 1 \
|1 + -----------|
| / pi\|
| sec|x - --||
\ \ 2 //
$$\frac{1 - \frac{\sqrt{2}}{\sec{\left(x + \frac{\pi}{4} \right)}}}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2}}$$
/ 2/x\\ ___ /x pi\
2*|1 - tan |-|| 2*\/ 2 *tan|- + --|
\ \2// \2 8 /
1 - --------------- + -------------------
2/x\ 2/x pi\
1 + tan |-| 1 + tan |- + --|
\2/ \2 8 /
-----------------------------------------
2
/ /x\ \
| 2*tan|-| |
| \2/ |
|1 + -----------|
| 2/x\|
| 1 + tan |-||
\ \2//
$$\frac{- \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1 + \frac{2 \sqrt{2} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}}$$
___
2 \/ 2
1 - ----------- + -----------
/pi \ / pi\
csc|-- - x| csc|x + --|
\2 / \ 4 /
-----------------------------
2
/ 1 \
|1 + ------|
\ csc(x)/
$$\frac{1 + \frac{\sqrt{2}}{\csc{\left(x + \frac{\pi}{4} \right)}} - \frac{2}{\csc{\left(- x + \frac{\pi}{2} \right)}}}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right)^{2}}$$
___ / pi\
1 - \/ 2 *cos|x + --|
\ 4 /
---------------------
2
/ / pi\\
|1 + cos|x - --||
\ \ 2 //
$$\frac{- \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} + 1}{\left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)^{2}}$$
/ pi\
cos|x - --|
\ 2 / (1 - cos(x))*cos(x)
--------------- - -------------------
/ pi\ 2
1 + cos|x - --| / / pi\\
\ 2 / |1 + cos|x - --||
\ \ 2 //
$$- \frac{\left(1 - \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)^{2}} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x - \frac{\pi}{2} \right)} + 1}$$
1
1 - ------
1 sec(x)
------------------- - --------------------
/ 1 \ 2
|1 + ------|*csc(x) / 1 \
\ csc(x)/ |1 + ------| *sec(x)
\ csc(x)/
$$\frac{1}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}} - \frac{1 - \frac{1}{\sec{\left(x \right)}}}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right)^{2} \sec{\left(x \right)}}$$
___ / pi\
1 - \/ 2 *cos|x + --|
\ 4 /
---------------------
2
(1 + sin(x))
$$\frac{- \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} + 1}{\left(\sin{\left(x \right)} + 1\right)^{2}}$$
___
2 \/ 2
1 - ------ + -----------
sec(x) / pi\
sec|x - --|
\ 4 /
------------------------
2
/ 1 \
|1 + -----------|
| / pi\|
| sec|x - --||
\ \ 2 //
$$\frac{1 + \frac{\sqrt{2}}{\sec{\left(x - \frac{\pi}{4} \right)}} - \frac{2}{\sec{\left(x \right)}}}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2}}$$
___ 2/x pi\ /x pi\
1 - 2*cos(x) + 2*\/ 2 *sin |- + --|*cot|- + --|
\2 8 / \2 8 /
-----------------------------------------------
2
(1 + sin(x))
$$\frac{2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 2 \cos{\left(x \right)} + 1}{\left(\sin{\left(x \right)} + 1\right)^{2}}$$
1
1 - ------
1 sec(x)
----------------------------- - -------------------------
/ 1 \ / pi\ 2
|1 + -----------|*sec|x - --| / 1 \
| / pi\| \ 2 / |1 + -----------| *sec(x)
| sec|x - --|| | / pi\|
\ \ 2 // | sec|x - --||
\ \ 2 //
$$- \frac{1 - \frac{1}{\sec{\left(x \right)}}}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2} \sec{\left(x \right)}} + \frac{1}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}}$$
___ / pi\
1 - 2*cos(x) + \/ 2 *sin|x + --|
\ 4 /
--------------------------------
2
(1 + sin(x))
$$\frac{\sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} - 2 \cos{\left(x \right)} + 1}{\left(\sin{\left(x \right)} + 1\right)^{2}}$$
___ / 2/x pi\\
\/ 2 *|-1 + cot |- + --||
\ \2 8 //
1 - -------------------------
2/x pi\
1 + cot |- + --|
\2 8 /
-----------------------------
2
/ /x\ \
| 2*cot|-| |
| \2/ |
|1 + -----------|
| 2/x\|
| 1 + cot |-||
\ \2//
$$\frac{- \frac{\sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} + 1}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}}$$
___ / 2/x pi\\
\/ 2 *|1 - tan |- + --||
\ \2 8 //
1 - ------------------------
2/x pi\
1 + tan |- + --|
\2 8 /
----------------------------
2
/ /x\ \
| 2*tan|-| |
| \2/ |
|1 + -----------|
| 2/x\|
| 1 + tan |-||
\ \2//
$$\frac{- \frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} + 1}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}}$$
___ / 2/x pi\\
\/ 2 *|1 - tan |- + --||
\ \2 8 //
1 - ------------------------
2/x pi\
1 + tan |- + --|
\2 8 /
----------------------------
2
(1 + sin(x))
$$\frac{- \frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} + 1}{\left(\sin{\left(x \right)} + 1\right)^{2}}$$
1
1 - -----------
/pi \
csc|-- - x|
1 \2 /
------------------- - -------------------------
/ 1 \ 2
|1 + ------|*csc(x) / 1 \ /pi \
\ csc(x)/ |1 + ------| *csc|-- - x|
\ csc(x)/ \2 /
$$\frac{1}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}} - \frac{1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right)^{2} \csc{\left(- x + \frac{\pi}{2} \right)}}$$
/ pi\ ___ / pi\
1 - 2*sin|x + --| + \/ 2 *sin|x + --|
\ 2 / \ 4 /
-------------------------------------
2
(1 + sin(x))
$$\frac{\sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} - 2 \sin{\left(x + \frac{\pi}{2} \right)} + 1}{\left(\sin{\left(x \right)} + 1\right)^{2}}$$
___
\/ 2
1 - ------------
/ pi\
csc|-x + --|
\ 4 /
----------------
2
/ 1 \
|1 + ------|
\ csc(x)/
$$\frac{1 - \frac{\sqrt{2}}{\csc{\left(- x + \frac{\pi}{4} \right)}}}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right)^{2}}$$
___ /x pi\
2*\/ 2 *tan|- + --|
\2 8 /
1 - 2*cos(x) + -------------------
2/x pi\
1 + tan |- + --|
\2 8 /
----------------------------------
2
(1 + sin(x))
$$\frac{- 2 \cos{\left(x \right)} + 1 + \frac{2 \sqrt{2} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}}{\left(\sin{\left(x \right)} + 1\right)^{2}}$$
___ 2/x pi\ / 2/x pi\\
1 - \/ 2 *sin |- + --|*|-1 + cot |- + --||
\2 8 / \ \2 8 //
------------------------------------------
2
(1 + sin(x))
$$\frac{- \sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}{\left(\sin{\left(x \right)} + 1\right)^{2}}$$
/ 2/x\\ ___ /x pi\
2*|-1 + cot |-|| 2*\/ 2 *cot|- + --|
\ \2// \2 8 /
1 - ---------------- + -------------------
2/x\ 2/x pi\
1 + cot |-| 1 + cot |- + --|
\2/ \2 8 /
------------------------------------------
2
/ /x\ \
| 2*cot|-| |
| \2/ |
|1 + -----------|
| 2/x\|
| 1 + cot |-||
\ \2//
$$\frac{- \frac{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1 + \frac{2 \sqrt{2} \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}}$$
/ 2/x\\
| 1 - tan |-||
/ 2/x\\ | \2/|
|1 - tan |-||*|1 - -----------|
/x\ \ \2// | 2/x\|
2*tan|-| | 1 + tan |-||
\2/ \ \2//
------------------------------- - --------------------------------
/ /x\ \ 2
| 2*tan|-| | / /x\ \
/ 2/x\\ | \2/ | | 2*tan|-| |
|1 + tan |-||*|1 + -----------| / 2/x\\ | \2/ |
\ \2// | 2/x\| |1 + tan |-||*|1 + -----------|
| 1 + tan |-|| \ \2// | 2/x\|
\ \2// | 1 + tan |-||
\ \2//
$$\frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ 2/x\\
| -1 + cot |-||
| \2/| / 2/x\\
|1 - ------------|*|-1 + cot |-||
/x\ | 2/x\ | \ \2//
2*cot|-| | 1 + cot |-| |
\2/ \ \2/ /
------------------------------- - ---------------------------------
/ /x\ \ 2
| 2*cot|-| | / /x\ \
/ 2/x\\ | \2/ | | 2*cot|-| |
|1 + cot |-||*|1 + -----------| / 2/x\\ | \2/ |
\ \2// | 2/x\| |1 + cot |-||*|1 + -----------|
| 1 + cot |-|| \ \2// | 2/x\|
\ \2// | 1 + cot |-||
\ \2//
$$\frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ / pi\\ / pi\
|1 - sin|x + --||*sin|x + --|
sin(x) \ \ 2 // \ 2 /
---------- - -----------------------------
1 + sin(x) 2
(1 + sin(x))
$$- \frac{\left(1 - \sin{\left(x + \frac{\pi}{2} \right)}\right) \sin{\left(x + \frac{\pi}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} + \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + 1}$$
___ / pi\
1 - 2*cos(x) + \/ 2 *cos|x - --|
\ 4 /
--------------------------------
2
/ / pi\\
|1 + cos|x - --||
\ \ 2 //
$$\frac{- 2 \cos{\left(x \right)} + \sqrt{2} \cos{\left(x - \frac{\pi}{4} \right)} + 1}{\left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)^{2}}$$
(1 - 2*cos(x) + sqrt(2)*cos(x - pi/4))/(1 + cos(x - pi/2))^2