Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(x/2)^3/(2*(2+cos(x/2))^2)+cos(x/2)*sin(x/2)/(2+cos(x/2))
¿Cómo vas a descomponer esta sin(x/2)^3/(2*(2+cos(x/2))^2)+cos(x/2)*sin(x/2)/(2+cos(x/2)) expresión en fracciones?
Solución
Ha introducido
[src]
3/x\ /x\ /x\
sin |-| cos|-|*sin|-|
\2/ \2/ \2/
--------------- + -------------
2 /x\
/ /x\\ 2 + cos|-|
2*|2 + cos|-|| \2/
\ \2//
$$\frac{\sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}{\cos{\left(\frac{x}{2} \right)} + 2} + \frac{\sin^{3}{\left(\frac{x}{2} \right)}}{2 \left(\cos{\left(\frac{x}{2} \right)} + 2\right)^{2}}$$
sin(x/2)^3/((2*(2 + cos(x/2))^2)) + (cos(x/2)*sin(x/2))/(2 + cos(x/2))
Simplificación general
[src]
/3 cos(x) /x\\ /x\
|- + ------ + 4*cos|-||*sin|-|
\2 2 \2// \2/
------------------------------
2
/ /x\\
2*|2 + cos|-||
\ \2//
$$\frac{\left(4 \cos{\left(\frac{x}{2} \right)} + \frac{\cos{\left(x \right)}}{2} + \frac{3}{2}\right) \sin{\left(\frac{x}{2} \right)}}{2 \left(\cos{\left(\frac{x}{2} \right)} + 2\right)^{2}}$$
(3/2 + cos(x)/2 + 4*cos(x/2))*sin(x/2)/(2*(2 + cos(x/2))^2)
Potencias
[src]
/ I*x -I*x \
3 | --- -----| / -I*x I*x\
/ -I*x I*x\ | 2 2 | | ----- ---|
| ----- ---| |e e | | 2 2 |
| 2 2 | I*|---- + ------|*\- e + e /
I*\- e + e / \ 2 2 /
----------------------- - -----------------------------------
2 / I*x -I*x \
/ I*x -I*x \ | --- -----|
| --- -----| | 2 2 |
| 2 2 | | e e |
| e e | 2*|2 + ---- + ------|
16*|2 + ---- + ------| \ 2 2 /
\ 2 2 /
$$- \frac{i \left(\frac{e^{\frac{i x}{2}}}{2} + \frac{e^{- \frac{i x}{2}}}{2}\right) \left(e^{\frac{i x}{2}} - e^{- \frac{i x}{2}}\right)}{2 \left(\frac{e^{\frac{i x}{2}}}{2} + 2 + \frac{e^{- \frac{i x}{2}}}{2}\right)} + \frac{i \left(e^{\frac{i x}{2}} - e^{- \frac{i x}{2}}\right)^{3}}{16 \left(\frac{e^{\frac{i x}{2}}}{2} + 2 + \frac{e^{- \frac{i x}{2}}}{2}\right)^{2}}$$
i*(-exp(-i*x/2) + exp(i*x/2))^3/(16*(2 + exp(i*x/2)/2 + exp(-i*x/2)/2)^2) - i*(exp(i*x/2)/2 + exp(-i*x/2)/2)*(-exp(-i*x/2) + exp(i*x/2))/(2*(2 + exp(i*x/2)/2 + exp(-i*x/2)/2))
Denominador racional
[src]
2
3/x\ / /x\\ / /x\\ /x\ /x\
sin |-|*|2 + cos|-|| + 2*|2 + cos|-|| *cos|-|*sin|-|
\2/ \ \2// \ \2// \2/ \2/
----------------------------------------------------
3
/ /x\\
2*|2 + cos|-||
\ \2//
$$\frac{2 \left(\cos{\left(\frac{x}{2} \right)} + 2\right)^{2} \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} + \left(\cos{\left(\frac{x}{2} \right)} + 2\right) \sin^{3}{\left(\frac{x}{2} \right)}}{2 \left(\cos{\left(\frac{x}{2} \right)} + 2\right)^{3}}$$
(sin(x/2)^3*(2 + cos(x/2)) + 2*(2 + cos(x/2))^2*cos(x/2)*sin(x/2))/(2*(2 + cos(x/2))^3)
Unión de expresiones racionales
[src]
/ 2/x\ / /x\\ /x\\ /x\
|sin |-| + 2*|2 + cos|-||*cos|-||*sin|-|
\ \2/ \ \2// \2// \2/
----------------------------------------
2
/ /x\\
2*|2 + cos|-||
\ \2//
$$\frac{\left(2 \left(\cos{\left(\frac{x}{2} \right)} + 2\right) \cos{\left(\frac{x}{2} \right)} + \sin^{2}{\left(\frac{x}{2} \right)}\right) \sin{\left(\frac{x}{2} \right)}}{2 \left(\cos{\left(\frac{x}{2} \right)} + 2\right)^{2}}$$
(sin(x/2)^2 + 2*(2 + cos(x/2))*cos(x/2))*sin(x/2)/(2*(2 + cos(x/2))^2)
Combinatoria
[src]
/ 2/x\ 2/x\ /x\\ /x\
|sin |-| + 2*cos |-| + 4*cos|-||*sin|-|
\ \2/ \2/ \2// \2/
---------------------------------------
2
/ /x\\
2*|2 + cos|-||
\ \2//
$$\frac{\left(\sin^{2}{\left(\frac{x}{2} \right)} + 2 \cos^{2}{\left(\frac{x}{2} \right)} + 4 \cos{\left(\frac{x}{2} \right)}\right) \sin{\left(\frac{x}{2} \right)}}{2 \left(\cos{\left(\frac{x}{2} \right)} + 2\right)^{2}}$$
(sin(x/2)^2 + 2*cos(x/2)^2 + 4*cos(x/2))*sin(x/2)/(2*(2 + cos(x/2))^2)
Respuesta numérica
[src]
0.125*sin(x/2)^3/(1 + 0.5*cos(x/2))^2 + cos(x/2)*sin(x/2)/(2.0 + cos(x/2))
0.125*sin(x/2)^3/(1 + 0.5*cos(x/2))^2 + cos(x/2)*sin(x/2)/(2.0 + cos(x/2))
Parte trigonométrica
[src]
1 1 5
------ + ---------- + --------
csc(x) /3*x\ /x\
8*csc|---| 8*csc|-|
\ 2 / \2/
------------------------------
2
/ 1 \
|2 + -----------|
| /pi x\|
| csc|-- - -||
\ \2 2//
$$\frac{\frac{1}{8 \csc{\left(\frac{3 x}{2} \right)}} + \frac{1}{\csc{\left(x \right)}} + \frac{5}{8 \csc{\left(\frac{x}{2} \right)}}}{\left(2 + \frac{1}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}\right)^{2}}$$
/x\ /3*x\ /x\
2*cot|-| cot|---| 5*cot|-|
\2/ \ 4 / \4/
----------- + ----------------- + ---------------
2/x\ / 2/3*x\\ / 2/x\\
1 + cot |-| 4*|1 + cot |---|| 4*|1 + cot |-||
\2/ \ \ 4 // \ \4//
-------------------------------------------------
2
/ 2/x\\
| -1 + cot |-||
| \4/|
|2 + ------------|
| 2/x\ |
| 1 + cot |-| |
\ \4/ /
$$\frac{\frac{\cot{\left(\frac{3 x}{4} \right)}}{4 \left(\cot^{2}{\left(\frac{3 x}{4} \right)} + 1\right)} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{5 \cot{\left(\frac{x}{4} \right)}}{4 \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} \right)} + 1} + 2\right)^{2}}$$
3/x pi\ /x\ /x pi\
cos |- - --| cos|-|*cos|- - --|
\2 2 / \2/ \2 2 /
--------------- + ------------------
2 /x\
/ /x\\ 2 + cos|-|
2*|2 + cos|-|| \2/
\ \2//
$$\frac{\cos{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)} + 2} + \frac{\cos^{3}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2 \left(\cos{\left(\frac{x}{2} \right)} + 2\right)^{2}}$$
1 1
----------------------- + --------------------------
2 / 1 \ /x\ /x\
/ 1 \ 3/x\ |2 + ------|*csc|-|*sec|-|
2*|2 + ------| *csc |-| | /x\| \2/ \2/
| /x\| \2/ | sec|-||
| sec|-|| \ \2//
\ \2//
$$\frac{1}{\left(2 + \frac{1}{\sec{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)} \sec{\left(\frac{x}{2} \right)}} + \frac{1}{2 \left(2 + \frac{1}{\sec{\left(\frac{x}{2} \right)}}\right)^{2} \csc^{3}{\left(\frac{x}{2} \right)}}$$
/3*x\ /x\
sin|---| 5*sin|-|
\ 2 / \2/
-------- + -------- + sin(x)
8 8
----------------------------
2
/ /pi x\\
|2 + sin|-- + -||
\ \2 2//
$$\frac{\frac{5 \sin{\left(\frac{x}{2} \right)}}{8} + \sin{\left(x \right)} + \frac{\sin{\left(\frac{3 x}{2} \right)}}{8}}{\left(\sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)} + 2\right)^{2}}$$
3/x\ / 2/x\\ /x\
4*tan |-| 2*|1 - tan |-||*tan|-|
\4/ \ \4// \4/
--------------------------------- + --------------------------------
2 / 2/x\\
/ 2/x\\ 2 | 1 - tan |-||
3 | 1 - tan |-|| / 2/x\\ | \4/|
/ 2/x\\ | \4/| |1 + tan |-|| *|2 + -----------|
|1 + tan |-|| *|2 + -----------| \ \4// | 2/x\|
\ \4// | 2/x\| | 1 + tan |-||
| 1 + tan |-|| \ \4//
\ \4//
$$\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right) \tan{\left(\frac{x}{4} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1} + 2\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}} + \frac{4 \tan^{3}{\left(\frac{x}{4} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1} + 2\right)^{2} \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{3}}$$
/x\ /3*x\ /x\
2*tan|-| tan|---| 5*tan|-|
\2/ \ 4 / \4/
----------- + ----------------- + ---------------
2/x\ / 2/3*x\\ / 2/x\\
1 + tan |-| 4*|1 + tan |---|| 4*|1 + tan |-||
\2/ \ \ 4 // \ \4//
-------------------------------------------------
2
/ 2/x\\
| 1 - tan |-||
| \4/|
|2 + -----------|
| 2/x\|
| 1 + tan |-||
\ \4//
$$\frac{\frac{\tan{\left(\frac{3 x}{4} \right)}}{4 \left(\tan^{2}{\left(\frac{3 x}{4} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{5 \tan{\left(\frac{x}{4} \right)}}{4 \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1} + 2\right)^{2}}$$
1 1
---------------------------- + ------------------------------------
2 / 1 \ /x\ /pi x\
/ 1 \ 3/x\ |2 + -----------|*csc|-|*csc|-- - -|
2*|2 + -----------| *csc |-| | /pi x\| \2/ \2 2/
| /pi x\| \2/ | csc|-- - -||
| csc|-- - -|| \ \2 2//
\ \2 2//
$$\frac{1}{\left(2 + \frac{1}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{1}{2 \left(2 + \frac{1}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}\right)^{2} \csc^{3}{\left(\frac{x}{2} \right)}}$$
1 1
---------------------------- + -------------------------------
2 / 1 \ /x\ /x pi\
/ 1 \ 3/x pi\ |2 + ------|*sec|-|*sec|- - --|
2*|2 + ------| *sec |- - --| | /x\| \2/ \2 2 /
| /x\| \2 2 / | sec|-||
| sec|-|| \ \2//
\ \2//
$$\frac{1}{\left(2 + \frac{1}{\sec{\left(\frac{x}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{1}{2 \left(2 + \frac{1}{\sec{\left(\frac{x}{2} \right)}}\right)^{2} \sec^{3}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
3/x\ / 2/x\\ /x\
4*cot |-| 2*|-1 + cot |-||*cot|-|
\4/ \ \4// \4/
---------------------------------- + ---------------------------------
2 / 2/x\\
/ 2/x\\ 2 | -1 + cot |-||
3 | -1 + cot |-|| / 2/x\\ | \4/|
/ 2/x\\ | \4/| |1 + cot |-|| *|2 + ------------|
|1 + cot |-|| *|2 + ------------| \ \4// | 2/x\ |
\ \4// | 2/x\ | | 1 + cot |-| |
| 1 + cot |-| | \ \4/ /
\ \4/ /
$$\frac{2 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right) \cot{\left(\frac{x}{4} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} \right)} + 1} + 2\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}} + \frac{4 \cot^{3}{\left(\frac{x}{4} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} \right)} + 1} + 2\right)^{2} \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{3}}$$
1 1 5
----------- + ----------------- + -------------
/ pi\ / pi 3*x\ /x pi\
sec|x - --| 8*sec|- -- + ---| 8*sec|- - --|
\ 2 / \ 2 2 / \2 2 /
-----------------------------------------------
2
/ 1 \
|2 + ------|
| /x\|
| sec|-||
\ \2//
$$\frac{\frac{1}{8 \sec{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{5}{8 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}}{\left(2 + \frac{1}{\sec{\left(\frac{x}{2} \right)}}\right)^{2}}$$
3/x\ /x\ /pi x\
sin |-| sin|-|*sin|-- + -|
\2/ \2/ \2 2/
-------------------- + ------------------
2 /pi x\
/ /pi x\\ 2 + sin|-- + -|
2*|2 + sin|-- + -|| \2 2/
\ \2 2//
$$\frac{\sin{\left(\frac{x}{2} \right)} \sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)} + 2} + \frac{\sin^{3}{\left(\frac{x}{2} \right)}}{2 \left(\sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)} + 2\right)^{2}}$$
/3*x\ /x\
sin|---| 5*sin|-|
\ 2 / \2/
-------- + -------- + sin(x)
8 8
----------------------------
2
/ /x\\
|2 + cos|-||
\ \2//
$$\frac{\frac{5 \sin{\left(\frac{x}{2} \right)}}{8} + \sin{\left(x \right)} + \frac{\sin{\left(\frac{3 x}{2} \right)}}{8}}{\left(\cos{\left(\frac{x}{2} \right)} + 2\right)^{2}}$$
/ pi 3*x\ /x pi\
cos|- -- + ---| 5*cos|- - --|
\ 2 2 / \2 2 / / pi\
--------------- + ------------- + cos|x - --|
8 8 \ 2 /
---------------------------------------------
2
/ /x\\
|2 + cos|-||
\ \2//
$$\frac{\frac{5 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{8} + \cos{\left(x - \frac{\pi}{2} \right)} + \frac{\cos{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}}{8}}{\left(\cos{\left(\frac{x}{2} \right)} + 2\right)^{2}}$$
(cos(-pi/2 + 3*x/2)/8 + 5*cos(x/2 - pi/2)/8 + cos(x - pi/2))/(2 + cos(x/2))^2
Denominador común
[src]
3/x\ /x\ /x\ /x\
- sin |-| + 8*sin|-| + 4*cos|-|*sin|-|
\2/ \2/ \2/ \2/ /x\
- -------------------------------------- + sin|-|
2/x\ /x\ \2/
8 + 2*cos |-| + 8*cos|-|
\2/ \2/
$$- \frac{- \sin^{3}{\left(\frac{x}{2} \right)} + 4 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} + 8 \sin{\left(\frac{x}{2} \right)}}{2 \cos^{2}{\left(\frac{x}{2} \right)} + 8 \cos{\left(\frac{x}{2} \right)} + 8} + \sin{\left(\frac{x}{2} \right)}$$
-(-sin(x/2)^3 + 8*sin(x/2) + 4*cos(x/2)*sin(x/2))/(8 + 2*cos(x/2)^2 + 8*cos(x/2)) + sin(x/2)