Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos(x)^2*(-2+9*x^2)/(2*x-3*x^3)^2-2*cos(x)*sin(x)/(2*x-3*x^3)
¿Cómo vas a descomponer esta cos(x)^2*(-2+9*x^2)/(2*x-3*x^3)^2-2*cos(x)*sin(x)/(2*x-3*x^3) expresión en fracciones?
Solución
Ha introducido
[src]
2 / 2\
cos (x)*\-2 + 9*x / 2*cos(x)*sin(x)
------------------- - ---------------
2 3
/ 3\ 2*x - 3*x
\2*x - 3*x /
$$\frac{\left(9 x^{2} - 2\right) \cos^{2}{\left(x \right)}}{\left(- 3 x^{3} + 2 x\right)^{2}} - \frac{\sin{\left(x \right)} 2 \cos{\left(x \right)}}{- 3 x^{3} + 2 x}$$
(cos(x)^2*(-2 + 9*x^2))/(2*x - 3*x^3)^2 - (2*cos(x))*sin(x)/(2*x - 3*x^3)
Simplificación general
[src]
// 2\ / 2\ \
\\-2 + 9*x /*cos(x) + 2*x*\-2 + 3*x /*sin(x)/*cos(x)
----------------------------------------------------
2
2 / 2\
x *\-2 + 3*x /
$$\frac{\left(2 x \left(3 x^{2} - 2\right) \sin{\left(x \right)} + \left(9 x^{2} - 2\right) \cos{\left(x \right)}\right) \cos{\left(x \right)}}{x^{2} \left(3 x^{2} - 2\right)^{2}}$$
((-2 + 9*x^2)*cos(x) + 2*x*(-2 + 3*x^2)*sin(x))*cos(x)/(x^2*(-2 + 3*x^2)^2)
Respuesta numérica
[src]
0.111111111111111*cos(x)^2*(-2.0 + 9.0*x^2)/(-x^3 + 0.666666666666667*x)^2 - 2.0*cos(x)*sin(x)/(2.0*x - 3.0*x^3)
0.111111111111111*cos(x)^2*(-2.0 + 9.0*x^2)/(-x^3 + 0.666666666666667*x)^2 - 2.0*cos(x)*sin(x)/(2.0*x - 3.0*x^3)
Unión de expresiones racionales
[src]
// 2\ / 2\ \
\\-2 + 9*x /*cos(x) - 2*x*\2 - 3*x /*sin(x)/*cos(x)
---------------------------------------------------
2
2 / 2\
x *\2 - 3*x /
$$\frac{\left(- 2 x \left(2 - 3 x^{2}\right) \sin{\left(x \right)} + \left(9 x^{2} - 2\right) \cos{\left(x \right)}\right) \cos{\left(x \right)}}{x^{2} \left(2 - 3 x^{2}\right)^{2}}$$
((-2 + 9*x^2)*cos(x) - 2*x*(2 - 3*x^2)*sin(x))*cos(x)/(x^2*(2 - 3*x^2)^2)
Potencias
[src]
2 / 2\
cos (x)*\-2 + 9*x / 2*cos(x)*sin(x)
------------------- - ---------------
2 3
/ 3 \ - 3*x + 2*x
\- 3*x + 2*x/
$$\frac{\left(9 x^{2} - 2\right) \cos^{2}{\left(x \right)}}{\left(- 3 x^{3} + 2 x\right)^{2}} - \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{- 3 x^{3} + 2 x}$$
2
/ I*x -I*x\
|e e | / 2\
|---- + -----| *\-2 + 9*x / / -I*x I*x\ / I*x -I*x\
\ 2 2 / I*\- e + e /*\e + e /
--------------------------- + ---------------------------------
2 / 3 \
/ 3 \ 2*\- 3*x + 2*x/
\- 3*x + 2*x/
$$\frac{\left(9 x^{2} - 2\right) \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2}}{\left(- 3 x^{3} + 2 x\right)^{2}} + \frac{i \left(e^{i x} - e^{- i x}\right) \left(e^{i x} + e^{- i x}\right)}{2 \left(- 3 x^{3} + 2 x\right)}$$
(exp(i*x)/2 + exp(-i*x)/2)^2*(-2 + 9*x^2)/(-3*x^3 + 2*x)^2 + i*(-exp(-i*x) + exp(i*x))*(exp(i*x) + exp(-i*x))/(2*(-3*x^3 + 2*x))
Combinatoria
[src]
/ 3 2 \
\-2*cos(x) - 4*x*sin(x) + 6*x *sin(x) + 9*x *cos(x)/*cos(x)
-----------------------------------------------------------
2
2 / 2\
x *\-2 + 3*x /
$$\frac{\left(6 x^{3} \sin{\left(x \right)} + 9 x^{2} \cos{\left(x \right)} - 4 x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \cos{\left(x \right)}}{x^{2} \left(3 x^{2} - 2\right)^{2}}$$
(-2*cos(x) - 4*x*sin(x) + 6*x^3*sin(x) + 9*x^2*cos(x))*cos(x)/(x^2*(-2 + 3*x^2)^2)
Compilar la expresión
[src]
2 / 2\
cos (x)*\-2 + 9*x / 2*cos(x)*sin(x)
------------------- - ---------------
2 3
/ 3 \ - 3*x + 2*x
\- 3*x + 2*x/
$$\frac{\left(9 x^{2} - 2\right) \cos^{2}{\left(x \right)}}{\left(- 3 x^{3} + 2 x\right)^{2}} - \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{- 3 x^{3} + 2 x}$$
cos(x)^2*(-2 + 9*x^2)/(-3*x^3 + 2*x)^2 - 2*cos(x)*sin(x)/(-3*x^3 + 2*x)
Denominador común
[src]
2 2 2 3
- 2*cos (x) + 9*x *cos (x) - 4*x*cos(x)*sin(x) + 6*x *cos(x)*sin(x)
-------------------------------------------------------------------
4 2 6
- 12*x + 4*x + 9*x
$$\frac{6 x^{3} \sin{\left(x \right)} \cos{\left(x \right)} + 9 x^{2} \cos^{2}{\left(x \right)} - 4 x \sin{\left(x \right)} \cos{\left(x \right)} - 2 \cos^{2}{\left(x \right)}}{9 x^{6} - 12 x^{4} + 4 x^{2}}$$
(-2*cos(x)^2 + 9*x^2*cos(x)^2 - 4*x*cos(x)*sin(x) + 6*x^3*cos(x)*sin(x))/(-12*x^4 + 4*x^2 + 9*x^6)
Parte trigonométrica
[src]
2
/ 2/x\\ / 2\
|1 - tan |-|| *\-2 + 9*x /
2*tan(x) \ \2//
- ---------------------------- + ------------------------------
/ 2 \ / 3 \ 2 2
\1 + tan (x)/*\- 3*x + 2*x/ / 2/x\\ / 3 \
|1 + tan |-|| *\- 3*x + 2*x/
\ \2//
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(9 x^{2} - 2\right)}{\left(- 3 x^{3} + 2 x\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{2 \tan{\left(x \right)}}{\left(- 3 x^{3} + 2 x\right) \left(\tan^{2}{\left(x \right)} + 1\right)}$$
/ 2 \
| -1 + cot (x)| / 2\
|1 + ------------|*\-2 + 9*x /
| 2 |
\ 1 + cot (x) / 2*cot(x)
------------------------------ - ----------------------------
2 / 2 \ / 3 \
/ 3 \ \1 + cot (x)/*\- 3*x + 2*x/
2*\- 3*x + 2*x/
$$\frac{\left(9 x^{2} - 2\right) \left(\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} + 1\right)}{2 \left(- 3 x^{3} + 2 x\right)^{2}} - \frac{2 \cot{\left(x \right)}}{\left(- 3 x^{3} + 2 x\right) \left(\cot^{2}{\left(x \right)} + 1\right)}$$
2
1 -2 + 9*x
- ---------------------------- + -----------------------
/ 3 \ / pi\ 2
\- 3*x + 2*x/*sec|2*x - --| / 3 \ 2
\ 2 / \- 3*x + 2*x/ *sec (x)
$$\frac{9 x^{2} - 2}{\left(- 3 x^{3} + 2 x\right)^{2} \sec^{2}{\left(x \right)}} - \frac{1}{\left(- 3 x^{3} + 2 x\right) \sec{\left(2 x - \frac{\pi}{2} \right)}}$$
2 / 2\
cos (x)*\-2 + 9*x / 2*cos(x)*sin(x)
------------------- - ---------------
2 3
/ 3 \ - 3*x + 2*x
\- 3*x + 2*x/
$$\frac{\left(9 x^{2} - 2\right) \cos^{2}{\left(x \right)}}{\left(- 3 x^{3} + 2 x\right)^{2}} - \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{- 3 x^{3} + 2 x}$$
2/ pi\ / 2\ / pi\
sin |x + --|*\-2 + 9*x / 2*sin(x)*sin|x + --|
\ 2 / \ 2 /
------------------------ - --------------------
2 3
/ 3 \ - 3*x + 2*x
\- 3*x + 2*x/
$$\frac{\left(9 x^{2} - 2\right) \sin^{2}{\left(x + \frac{\pi}{2} \right)}}{\left(- 3 x^{3} + 2 x\right)^{2}} - \frac{2 \sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)}}{- 3 x^{3} + 2 x}$$
2
-2 + 9*x 2
---------------------------- - ---------------------------------
2 / 3 \ /pi \
/ 3 \ 2/pi \ \- 3*x + 2*x/*csc(x)*csc|-- - x|
\- 3*x + 2*x/ *csc |-- - x| \2 /
\2 /
$$\frac{9 x^{2} - 2}{\left(- 3 x^{3} + 2 x\right)^{2} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - \frac{2}{\left(- 3 x^{3} + 2 x\right) \csc{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}$$
2
/ 2/x\\ / 2\ / 2/x\\ /x\
|1 - tan |-|| *\-2 + 9*x / 4*|1 - tan |-||*tan|-|
\ \2// \ \2// \2/
------------------------------ - -----------------------------
2 2 2
/ 2/x\\ / 3 \ / 2/x\\ / 3 \
|1 + tan |-|| *\- 3*x + 2*x/ |1 + tan |-|| *\- 3*x + 2*x/
\ \2// \ \2//
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(9 x^{2} - 2\right)}{\left(- 3 x^{3} + 2 x\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{4 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(\frac{x}{2} \right)}}{\left(- 3 x^{3} + 2 x\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
2
1 -2 + 9*x
- ----------------------- + ----------------------------
/ 3 \ 2
\- 3*x + 2*x/*csc(2*x) / 3 \ 2/pi \
\- 3*x + 2*x/ *csc |-- - x|
\2 /
$$\frac{9 x^{2} - 2}{\left(- 3 x^{3} + 2 x\right)^{2} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\left(- 3 x^{3} + 2 x\right) \csc{\left(2 x \right)}}$$
/ 2 \
| 1 - tan (x)| / 2\
|1 + -----------|*\-2 + 9*x /
| 2 |
\ 1 + tan (x)/ 2*tan(x)
----------------------------- - ----------------------------
2 / 2 \ / 3 \
/ 3 \ \1 + tan (x)/*\- 3*x + 2*x/
2*\- 3*x + 2*x/
$$\frac{\left(9 x^{2} - 2\right) \left(\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} + 1\right)}{2 \left(- 3 x^{3} + 2 x\right)^{2}} - \frac{2 \tan{\left(x \right)}}{\left(- 3 x^{3} + 2 x\right) \left(\tan^{2}{\left(x \right)} + 1\right)}$$
2 / 2\
sin(2*x) cos (x)*\-2 + 9*x /
- ---------- + -------------------
3 2
2*x - 3*x / 3\
\2*x - 3*x /
$$\frac{\left(9 x^{2} - 2\right) \cos^{2}{\left(x \right)}}{\left(- 3 x^{3} + 2 x\right)^{2}} - \frac{\sin{\left(2 x \right)}}{- 3 x^{3} + 2 x}$$
2
/ 2/x\\ / 2\ / 2/x\\ /x\
|-1 + cot |-|| *\-2 + 9*x / 4*|-1 + cot |-||*cot|-|
\ \2// \ \2// \2/
------------------------------ - -----------------------------
2 2 2
/ 2/x\\ / 3 \ / 2/x\\ / 3 \
|1 + cot |-|| *\- 3*x + 2*x/ |1 + cot |-|| *\- 3*x + 2*x/
\ \2// \ \2//
$$\frac{\left(9 x^{2} - 2\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(- 3 x^{3} + 2 x\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{4 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \cot{\left(\frac{x}{2} \right)}}{\left(- 3 x^{3} + 2 x\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
/ 1 \ / 2\
|1 + -------------|*\-2 + 9*x /
| /pi \|
| csc|-- - 2*x||
1 \ \2 //
- ----------------------- + -------------------------------
/ 3 \ 2
\- 3*x + 2*x/*csc(2*x) / 3 \
2*\- 3*x + 2*x/
$$\frac{\left(1 + \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}\right) \left(9 x^{2} - 2\right)}{2 \left(- 3 x^{3} + 2 x\right)^{2}} - \frac{1}{\left(- 3 x^{3} + 2 x\right) \csc{\left(2 x \right)}}$$
2
-2 + 9*x 2
----------------------- - ----------------------------
2 / 3 \
/ 3 \ 2 \- 3*x + 2*x/*csc(x)*sec(x)
\- 3*x + 2*x/ *sec (x)
$$\frac{9 x^{2} - 2}{\left(- 3 x^{3} + 2 x\right)^{2} \sec^{2}{\left(x \right)}} - \frac{2}{\left(- 3 x^{3} + 2 x\right) \csc{\left(x \right)} \sec{\left(x \right)}}$$
/ pi\
2 / 2\ 2*cos(x)*cos|x - --|
cos (x)*\-2 + 9*x / \ 2 /
------------------- - --------------------
2 3
/ 3 \ - 3*x + 2*x
\- 3*x + 2*x/
$$\frac{\left(9 x^{2} - 2\right) \cos^{2}{\left(x \right)}}{\left(- 3 x^{3} + 2 x\right)^{2}} - \frac{2 \cos{\left(x \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{- 3 x^{3} + 2 x}$$
2 / 2\
sin(2*x) cos (x)*\-2 + 9*x /
- ------------ + -------------------
3 2
- 3*x + 2*x / 3 \
\- 3*x + 2*x/
$$\frac{\left(9 x^{2} - 2\right) \cos^{2}{\left(x \right)}}{\left(- 3 x^{3} + 2 x\right)^{2}} - \frac{\sin{\left(2 x \right)}}{- 3 x^{3} + 2 x}$$
/ pi\
cos|2*x - --| 2 / 2\
\ 2 / cos (x)*\-2 + 9*x /
- ------------- + -------------------
3 2
- 3*x + 2*x / 3 \
\- 3*x + 2*x/
$$\frac{\left(9 x^{2} - 2\right) \cos^{2}{\left(x \right)}}{\left(- 3 x^{3} + 2 x\right)^{2}} - \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{- 3 x^{3} + 2 x}$$
/ 1 \ / 2\
|1 + --------|*\-2 + 9*x /
1 \ sec(2*x)/
- ---------------------------- + --------------------------
/ 3 \ / pi\ 2
\- 3*x + 2*x/*sec|2*x - --| / 3 \
\ 2 / 2*\- 3*x + 2*x/
$$\frac{\left(1 + \frac{1}{\sec{\left(2 x \right)}}\right) \left(9 x^{2} - 2\right)}{2 \left(- 3 x^{3} + 2 x\right)^{2}} - \frac{1}{\left(- 3 x^{3} + 2 x\right) \sec{\left(2 x - \frac{\pi}{2} \right)}}$$
2/ pi\ / 2\
sin |x + --|*\-2 + 9*x /
sin(2*x) \ 2 /
- ------------ + ------------------------
3 2
- 3*x + 2*x / 3 \
\- 3*x + 2*x/
$$\frac{\left(9 x^{2} - 2\right) \sin^{2}{\left(x + \frac{\pi}{2} \right)}}{\left(- 3 x^{3} + 2 x\right)^{2}} - \frac{\sin{\left(2 x \right)}}{- 3 x^{3} + 2 x}$$
2
-2 + 9*x 2
----------------------- - ---------------------------------
2 / 3 \ / pi\
/ 3 \ 2 \- 3*x + 2*x/*sec(x)*sec|x - --|
\- 3*x + 2*x/ *sec (x) \ 2 /
$$\frac{9 x^{2} - 2}{\left(- 3 x^{3} + 2 x\right)^{2} \sec^{2}{\left(x \right)}} - \frac{2}{\left(- 3 x^{3} + 2 x\right) \sec{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}$$
/ pi\
cos|2*x - --| / 2\
\ 2 / (1 + cos(2*x))*\-2 + 9*x /
- ------------- + --------------------------
3 2
- 3*x + 2*x / 3 \
2*\- 3*x + 2*x/
$$\frac{\left(9 x^{2} - 2\right) \left(\cos{\left(2 x \right)} + 1\right)}{2 \left(- 3 x^{3} + 2 x\right)^{2}} - \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{- 3 x^{3} + 2 x}$$
/ /pi \\ / 2\
|1 + sin|-- + 2*x||*\-2 + 9*x /
sin(2*x) \ \2 //
- ------------ + -------------------------------
3 2
- 3*x + 2*x / 3 \
2*\- 3*x + 2*x/
$$\frac{\left(9 x^{2} - 2\right) \left(\sin{\left(2 x + \frac{\pi}{2} \right)} + 1\right)}{2 \left(- 3 x^{3} + 2 x\right)^{2}} - \frac{\sin{\left(2 x \right)}}{- 3 x^{3} + 2 x}$$
/ 2\
sin(2*x) (1 + cos(2*x))*\-2 + 9*x /
- ------------ + --------------------------
3 2
- 3*x + 2*x / 3 \
2*\- 3*x + 2*x/
$$\frac{\left(9 x^{2} - 2\right) \left(\cos{\left(2 x \right)} + 1\right)}{2 \left(- 3 x^{3} + 2 x\right)^{2}} - \frac{\sin{\left(2 x \right)}}{- 3 x^{3} + 2 x}$$
2
/ 2/x\\ / 2\
|-1 + cot |-|| *\-2 + 9*x /
2*cot(x) \ \2//
- ---------------------------- + ------------------------------
/ 2 \ / 3 \ 2 2
\1 + cot (x)/*\- 3*x + 2*x/ / 2/x\\ / 3 \
|1 + cot |-|| *\- 3*x + 2*x/
\ \2//
$$\frac{\left(9 x^{2} - 2\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(- 3 x^{3} + 2 x\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{2 \cot{\left(x \right)}}{\left(- 3 x^{3} + 2 x\right) \left(\cot^{2}{\left(x \right)} + 1\right)}$$
-2*cot(x)/((1 + cot(x)^2)*(-3*x^3 + 2*x)) + (-1 + cot(x/2)^2)^2*(-2 + 9*x^2)/((1 + cot(x/2)^2)^2*(-3*x^3 + 2*x)^2)
Denominador racional
[src]
2
2 / 2\ / 3 \ / 3 \
cos (x)*\-2 + 9*x /*\- 3*x + 2*x/ - 2*\- 3*x + 2*x/ *cos(x)*sin(x)
--------------------------------------------------------------------
3
/ 3 \
\- 3*x + 2*x/
$$\frac{\left(9 x^{2} - 2\right) \left(- 3 x^{3} + 2 x\right) \cos^{2}{\left(x \right)} - 2 \left(- 3 x^{3} + 2 x\right)^{2} \sin{\left(x \right)} \cos{\left(x \right)}}{\left(- 3 x^{3} + 2 x\right)^{3}}$$
(cos(x)^2*(-2 + 9*x^2)*(-3*x^3 + 2*x) - 2*(-3*x^3 + 2*x)^2*cos(x)*sin(x))/(-3*x^3 + 2*x)^3