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