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