Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos(x)/(1-sin(2*x))+2*cos(2*x)*sin(x)/(1-sin(2*x))^2
¿Cómo vas a descomponer esta cos(x)/(1-sin(2*x))+2*cos(2*x)*sin(x)/(1-sin(2*x))^2 expresión en fracciones?
Solución
Ha introducido
[src]
cos(x) 2*cos(2*x)*sin(x)
------------ + -----------------
1 - sin(2*x) 2
(1 - sin(2*x))
$$\frac{\sin{\left(x \right)} 2 \cos{\left(2 x \right)}}{\left(1 - \sin{\left(2 x \right)}\right)^{2}} + \frac{\cos{\left(x \right)}}{1 - \sin{\left(2 x \right)}}$$
cos(x)/(1 - sin(2*x)) + ((2*cos(2*x))*sin(x))/(1 - sin(2*x))^2
Simplificación general
[src]
3
- 2*sin (x) + cos(x)
--------------------
2
(-1 + sin(2*x))
$$\frac{- 2 \sin^{3}{\left(x \right)} + \cos{\left(x \right)}}{\left(\sin{\left(2 x \right)} - 1\right)^{2}}$$
(-2*sin(x)^3 + cos(x))/(-1 + sin(2*x))^2
Denominador racional
[src]
2
(1 - sin(2*x)) *cos(x) + 2*(1 - sin(2*x))*cos(2*x)*sin(x)
---------------------------------------------------------
3
(1 - sin(2*x))
$$\frac{\left(1 - \sin{\left(2 x \right)}\right)^{2} \cos{\left(x \right)} + 2 \left(1 - \sin{\left(2 x \right)}\right) \sin{\left(x \right)} \cos{\left(2 x \right)}}{\left(1 - \sin{\left(2 x \right)}\right)^{3}}$$
((1 - sin(2*x))^2*cos(x) + 2*(1 - sin(2*x))*cos(2*x)*sin(x))/(1 - sin(2*x))^3
Combinatoria
[src]
-cos(x)*sin(2*x) + 2*cos(2*x)*sin(x) + cos(x)
---------------------------------------------
2
(-1 + sin(2*x))
$$\frac{2 \sin{\left(x \right)} \cos{\left(2 x \right)} - \sin{\left(2 x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}}{\left(\sin{\left(2 x \right)} - 1\right)^{2}}$$
(-cos(x)*sin(2*x) + 2*cos(2*x)*sin(x) + cos(x))/(-1 + sin(2*x))^2
Denominador común
[src]
-cos(x)*sin(2*x) + 2*cos(2*x)*sin(x) + cos(x)
---------------------------------------------
2
1 + sin (2*x) - 2*sin(2*x)
$$\frac{2 \sin{\left(x \right)} \cos{\left(2 x \right)} - \sin{\left(2 x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}}{\sin^{2}{\left(2 x \right)} - 2 \sin{\left(2 x \right)} + 1}$$
(-cos(x)*sin(2*x) + 2*cos(2*x)*sin(x) + cos(x))/(1 + sin(2*x)^2 - 2*sin(2*x))
Unión de expresiones racionales
[src]
(1 - sin(2*x))*cos(x) + 2*cos(2*x)*sin(x)
-----------------------------------------
2
(1 - sin(2*x))
$$\frac{\left(1 - \sin{\left(2 x \right)}\right) \cos{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(2 x \right)}}{\left(1 - \sin{\left(2 x \right)}\right)^{2}}$$
((1 - sin(2*x))*cos(x) + 2*cos(2*x)*sin(x))/(1 - sin(2*x))^2
Potencias
[src]
I*x -I*x
e e
---- + ----- / -I*x I*x\ / -2*I*x 2*I*x\
2 2 I*\- e + e /*\e + e /
-------------------------- - -------------------------------------
/ -2*I*x 2*I*x\ 2
I*\- e + e / / / -2*I*x 2*I*x\\
1 + ---------------------- | I*\- e + e /|
2 2*|1 + ----------------------|
\ 2 /
$$\frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{\frac{i \left(e^{2 i x} - e^{- 2 i x}\right)}{2} + 1} - \frac{i \left(e^{i x} - e^{- i x}\right) \left(e^{2 i x} + e^{- 2 i x}\right)}{2 \left(\frac{i \left(e^{2 i x} - e^{- 2 i x}\right)}{2} + 1\right)^{2}}$$
cos(x) 2*cos(2*x)*sin(x)
------------ + -----------------
1 - sin(2*x) 2
(1 - sin(2*x))
$$\frac{\cos{\left(x \right)}}{1 - \sin{\left(2 x \right)}} + \frac{2 \sin{\left(x \right)} \cos{\left(2 x \right)}}{\left(1 - \sin{\left(2 x \right)}\right)^{2}}$$
cos(x)/(1 - sin(2*x)) + 2*cos(2*x)*sin(x)/(1 - sin(2*x))^2
Compilar la expresión
[src]
cos(x) 2*cos(2*x)*sin(x)
------------ + -----------------
1 - sin(2*x) 2
(1 - sin(2*x))
$$\frac{\cos{\left(x \right)}}{1 - \sin{\left(2 x \right)}} + \frac{2 \sin{\left(x \right)} \cos{\left(2 x \right)}}{\left(1 - \sin{\left(2 x \right)}\right)^{2}}$$
cos(x)/(1 - sin(2*x)) + 2*cos(2*x)*sin(x)/(1 - sin(2*x))^2
Parte trigonométrica
[src]
1 2
-------------------------- + -----------------------------------------
/ 1 \ 2
|1 - -------------|*sec(x) / 1 \ / pi\
| / pi\| |1 - -------------| *sec(2*x)*sec|x - --|
| sec|2*x - --|| | / pi\| \ 2 /
\ \ 2 // | sec|2*x - --||
\ \ 2 //
$$\frac{1}{\left(1 - \frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}}\right) \sec{\left(x \right)}} + \frac{2}{\left(1 - \frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}}\right)^{2} \sec{\left(2 x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}$$
3/ pi\
- 2*cos |x - --| + cos(x)
\ 2 /
----------------------------
2
/ / pi\\
|-1 + 2*cos(x)*cos|x - --||
\ \ 2 //
$$\frac{\cos{\left(x \right)} - 2 \cos^{3}{\left(x - \frac{\pi}{2} \right)}}{\left(2 \cos{\left(x \right)} \cos{\left(x - \frac{\pi}{2} \right)} - 1\right)^{2}}$$
2/x\ / 2 \ /x\
-1 + cot |-| 4*\-1 + cot (x)/*cot|-|
\2/ \2/
------------------------------- + ----------------------------------------------
/ 2/x\\ / 2*cot(x) \ 2
|1 + cot |-||*|1 - -----------| / 2 \ / 2/x\\ / 2*cot(x) \
\ \2// | 2 | \1 + cot (x)/*|1 + cot |-||*|1 - -----------|
\ 1 + cot (x)/ \ \2// | 2 |
\ 1 + cot (x)/
$$\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\left(1 - \frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{4 \left(\cot^{2}{\left(x \right)} - 1\right) \cot{\left(\frac{x}{2} \right)}}{\left(1 - \frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}$$
2/x\ 3/x\
1 - tan |-| 16*tan |-|
\2/ \2/
----------- - --------------
2/x\ 3
1 + tan |-| / 2/x\\
\2/ |1 + tan |-||
\ \2//
------------------------------
2
/ / 2/x\\ /x\\
| 4*|1 - tan |-||*tan|-||
| \ \2// \2/|
|-1 + ----------------------|
| 2 |
| / 2/x\\ |
| |1 + tan |-|| |
\ \ \2// /
$$\frac{\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{16 \tan^{3}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}}{\left(\frac{4 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - 1\right)^{2}}$$
3 / pi\
- 2*sin (x) + sin|x + --|
\ 2 /
----------------------------
2
/ / pi\\
|-1 + 2*sin(x)*sin|x + --||
\ \ 2 //
$$\frac{- 2 \sin^{3}{\left(x \right)} + \sin{\left(x + \frac{\pi}{2} \right)}}{\left(2 \sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)} - 1\right)^{2}}$$
/ pi\ / pi\
cos|3*x - --| 3*cos|x - --|
\ 2 / \ 2 /
------------- - ------------- + cos(x)
2 2
--------------------------------------
2
/ / pi\\
|-1 + cos|2*x - --||
\ \ 2 //
$$\frac{\cos{\left(x \right)} - \frac{3 \cos{\left(x - \frac{\pi}{2} \right)}}{2} + \frac{\cos{\left(3 x - \frac{\pi}{2} \right)}}{2}}{\left(\cos{\left(2 x - \frac{\pi}{2} \right)} - 1\right)^{2}}$$
1 2
-------------------------- + ------------------------------------
/ 1 \ /pi \ 2
|1 - --------|*csc|-- - x| / 1 \ /pi \
\ csc(2*x)/ \2 / |1 - --------| *csc(x)*csc|-- - 2*x|
\ csc(2*x)/ \2 /
$$\frac{1}{\left(1 - \frac{1}{\csc{\left(2 x \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{2}{\left(1 - \frac{1}{\csc{\left(2 x \right)}}\right)^{2} \csc{\left(x \right)} \csc{\left(- 2 x + \frac{\pi}{2} \right)}}$$
2/x\ /3*x\ /x\
1 - tan |-| tan|---| 3*tan|-|
\2/ \ 2 / \2/
----------- + ------------- - -----------
2/x\ 2/3*x\ 2/x\
1 + tan |-| 1 + tan |---| 1 + tan |-|
\2/ \ 2 / \2/
-----------------------------------------
2
/ 2*tan(x) \
|-1 + -----------|
| 2 |
\ 1 + tan (x)/
$$\frac{\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{\tan{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1} - \frac{3 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}}{\left(-1 + \frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}\right)^{2}}$$
2/x\ / 2 \ /x\
1 - tan |-| 4*\1 - tan (x)/*tan|-|
\2/ \2/
------------------------------- + ----------------------------------------------
/ 2/x\\ / 2*tan(x) \ 2
|1 + tan |-||*|1 - -----------| / 2 \ / 2/x\\ / 2*tan(x) \
\ \2// | 2 | \1 + tan (x)/*|1 + tan |-||*|1 - -----------|
\ 1 + tan (x)/ \ \2// | 2 |
\ 1 + tan (x)/
$$\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\left(1 - \frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{4 \left(1 - \tan^{2}{\left(x \right)}\right) \tan{\left(\frac{x}{2} \right)}}{\left(1 - \frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)}$$
sin(3*x) 3*sin(x)
-------- - -------- + cos(x)
2 2
----------------------------
2
(-1 + sin(2*x))
$$\frac{- \frac{3 \sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{2} + \cos{\left(x \right)}}{\left(\sin{\left(2 x \right)} - 1\right)^{2}}$$
2/x\ 3/x\
-1 + cot |-| 16*cot |-|
\2/ \2/
------------ - --------------
2/x\ 3
1 + cot |-| / 2/x\\
\2/ |1 + cot |-||
\ \2//
-------------------------------
2
/ / 2/x\\ /x\\
| 4*|-1 + cot |-||*cot|-||
| \ \2// \2/|
|-1 + -----------------------|
| 2 |
| / 2/x\\ |
| |1 + cot |-|| |
\ \ \2// /
$$\frac{\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{16 \cot^{3}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}}{\left(\frac{4 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \cot{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - 1\right)^{2}}$$
1 1 3
------ + --------------- - -------------
sec(x) / pi\ / pi\
2*sec|3*x - --| 2*sec|x - --|
\ 2 / \ 2 /
----------------------------------------
2
/ 1 \
|-1 + -------------|
| / pi\|
| sec|2*x - --||
\ \ 2 //
$$\frac{\frac{1}{2 \sec{\left(3 x - \frac{\pi}{2} \right)}} - \frac{3}{2 \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(x \right)}}}{\left(-1 + \frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}}\right)^{2}}$$
1 2
------ - ------------
sec(x) 3/ pi\
sec |x - --|
\ 2 /
--------------------------
2
/ 2 \
|-1 + ------------------|
| / pi\|
| sec(x)*sec|x - --||
\ \ 2 //
$$\frac{- \frac{2}{\sec^{3}{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(x \right)}}}{\left(-1 + \frac{2}{\sec{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2}}$$
cos(x) 2*cos(2*x)*sin(x)
------------ + -----------------
1 - sin(2*x) 2
(1 - sin(2*x))
$$\frac{\cos{\left(x \right)}}{1 - \sin{\left(2 x \right)}} + \frac{2 \sin{\left(x \right)} \cos{\left(2 x \right)}}{\left(1 - \sin{\left(2 x \right)}\right)^{2}}$$
/ pi\ /pi \
sin|x + --| 2*sin(x)*sin|-- + 2*x|
\ 2 / \2 /
------------ + ----------------------
1 - sin(2*x) 2
(1 - sin(2*x))
$$\frac{\sin{\left(x + \frac{\pi}{2} \right)}}{1 - \sin{\left(2 x \right)}} + \frac{2 \sin{\left(x \right)} \sin{\left(2 x + \frac{\pi}{2} \right)}}{\left(1 - \sin{\left(2 x \right)}\right)^{2}}$$
1 2
----------- - -------
/pi \ 3
csc|-- - x| csc (x)
\2 /
--------------------------
2
/ 2 \
|-1 + ------------------|
| /pi \|
| csc(x)*csc|-- - x||
\ \2 //
$$\frac{\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{2}{\csc^{3}{\left(x \right)}}}{\left(-1 + \frac{2}{\csc{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2}}$$
3
- 2*sin (x) + cos(x)
--------------------
2
(-1 + sin(2*x))
$$\frac{- 2 \sin^{3}{\left(x \right)} + \cos{\left(x \right)}}{\left(\sin{\left(2 x \right)} - 1\right)^{2}}$$
/ pi\
2*cos(2*x)*cos|x - --|
cos(x) \ 2 /
----------------- + ----------------------
/ pi\ 2
1 - cos|2*x - --| / / pi\\
\ 2 / |1 - cos|2*x - --||
\ \ 2 //
$$\frac{\cos{\left(x \right)}}{1 - \cos{\left(2 x - \frac{\pi}{2} \right)}} + \frac{2 \cos{\left(2 x \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{\left(1 - \cos{\left(2 x - \frac{\pi}{2} \right)}\right)^{2}}$$
1 1 3
----------- + ---------- - --------
/pi \ 2*csc(3*x) 2*csc(x)
csc|-- - x|
\2 /
-----------------------------------
2
/ 1 \
|-1 + --------|
\ csc(2*x)/
$$\frac{\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{2 \csc{\left(3 x \right)}} - \frac{3}{2 \csc{\left(x \right)}}}{\left(-1 + \frac{1}{\csc{\left(2 x \right)}}\right)^{2}}$$
2/x\ /3*x\ /x\
-1 + cot |-| cot|---| 3*cot|-|
\2/ \ 2 / \2/
------------ + ------------- - -----------
2/x\ 2/3*x\ 2/x\
1 + cot |-| 1 + cot |---| 1 + cot |-|
\2/ \ 2 / \2/
------------------------------------------
2
/ 2*cot(x) \
|-1 + -----------|
| 2 |
\ 1 + cot (x)/
$$\frac{\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{\cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} - \frac{3 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}}{\left(-1 + \frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)^{2}}$$
1 2
--------------------- + -------------------------------
/ 1 \ 2
|1 - --------|*sec(x) / 1 \
\ csc(2*x)/ |1 - --------| *csc(x)*sec(2*x)
\ csc(2*x)/
$$\frac{1}{\left(1 - \frac{1}{\csc{\left(2 x \right)}}\right) \sec{\left(x \right)}} + \frac{2}{\left(1 - \frac{1}{\csc{\left(2 x \right)}}\right)^{2} \csc{\left(x \right)} \sec{\left(2 x \right)}}$$
sin(3*x) 3*sin(x) / pi\
-------- - -------- + sin|x + --|
2 2 \ 2 /
---------------------------------
2
(-1 + sin(2*x))
$$\frac{- \frac{3 \sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{2} + \sin{\left(x + \frac{\pi}{2} \right)}}{\left(\sin{\left(2 x \right)} - 1\right)^{2}}$$
-sin(3*x) + 3*sin(x)
- -------------------- + cos(x)
2
-------------------------------
2
(-1 + sin(2*x))
$$\frac{- \frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{2} + \cos{\left(x \right)}}{\left(\sin{\left(2 x \right)} - 1\right)^{2}}$$
3
- 2*sin (x) + cos(x)
-----------------------
2
(-1 + 2*cos(x)*sin(x))
$$\frac{- 2 \sin^{3}{\left(x \right)} + \cos{\left(x \right)}}{\left(2 \sin{\left(x \right)} \cos{\left(x \right)} - 1\right)^{2}}$$
(-2*sin(x)^3 + cos(x))/(-1 + 2*cos(x)*sin(x))^2
Respuesta numérica
[src]
cos(x)/(1.0 - sin(2*x)) + 2.0*cos(2*x)*sin(x)/(1.0 - sin(2*x))^2
cos(x)/(1.0 - sin(2*x)) + 2.0*cos(2*x)*sin(x)/(1.0 - sin(2*x))^2