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