Sr Examen
¿Cómo vas a descomponer esta sin(2x)/(16*(sin^2x-cos^2x)) expresión en fracciones?
Solución
Ha introducido
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sin(2*x) ---------------------- / 2 2 \ 16*\sin (x) - cos (x)/
$$\frac{\sin{\left(2 x \right)}}{16 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}$$
sin(2*x)/((16*(sin(x)^2 - cos(x)^2)))
Simplificación general
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-tan(2*x)
----------
16
$$- \frac{\tan{\left(2 x \right)}}{16}$$
-tan(2*x)/16
Respuesta numérica
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sin(2*x)/(16.0*sin(x)^2 - 16.0*cos(x)^2)
sin(2*x)/(16.0*sin(x)^2 - 16.0*cos(x)^2)
Unión de expresiones racionales
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sin(2*x)
-------------------------
2 2
- 16*cos (x) + 16*sin (x)
$$\frac{\sin{\left(2 x \right)}}{16 \sin^{2}{\left(x \right)} - 16 \cos^{2}{\left(x \right)}}$$
sin(2*x)/(-16*cos(x)^2 + 16*sin(x)^2)
Potencias
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sin(2*x)
-------------------------
2 2
- 16*cos (x) + 16*sin (x)
$$\frac{\sin{\left(2 x \right)}}{16 \sin^{2}{\left(x \right)} - 16 \cos^{2}{\left(x \right)}}$$
/ -2*I*x 2*I*x\
-I*\- e + e /
----------------------------------------------
/ 2 \
| / I*x -I*x\ 2|
| |e e | / -I*x I*x\ |
2*|- 16*|---- + -----| - 4*\- e + e / |
\ \ 2 2 / /
$$- \frac{i \left(e^{2 i x} - e^{- 2 i x}\right)}{2 \left(- 16 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} - 4 \left(e^{i x} - e^{- i x}\right)^{2}\right)}$$
-i*(-exp(-2*i*x) + exp(2*i*x))/(2*(-16*(exp(i*x)/2 + exp(-i*x)/2)^2 - 4*(-exp(-i*x) + exp(i*x))^2))
Denominador común
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-sin(2*x)
-------------------------
2 2
- 16*sin (x) + 16*cos (x)
$$- \frac{\sin{\left(2 x \right)}}{- 16 \sin^{2}{\left(x \right)} + 16 \cos^{2}{\left(x \right)}}$$
-sin(2*x)/(-16*sin(x)^2 + 16*cos(x)^2)
Compilar la expresión
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sin(2*x)
-------------------------
2 2
- 16*cos (x) + 16*sin (x)
$$\frac{\sin{\left(2 x \right)}}{16 \sin^{2}{\left(x \right)} - 16 \cos^{2}{\left(x \right)}}$$
sin(2*x)/(-16*cos(x)^2 + 16*sin(x)^2)
Combinatoria
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-sin(2*x) --------------------------------------- 16*(-sin(x) + cos(x))*(cos(x) + sin(x))
$$- \frac{\sin{\left(2 x \right)}}{16 \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}$$
-sin(2*x)/(16*(-sin(x) + cos(x))*(cos(x) + sin(x)))
Denominador racional
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sin(2*x)
-------------------------
2 2
- 16*cos (x) + 16*sin (x)
$$\frac{\sin{\left(2 x \right)}}{16 \sin^{2}{\left(x \right)} - 16 \cos^{2}{\left(x \right)}}$$
sin(2*x)/(-16*cos(x)^2 + 16*sin(x)^2)
Parte trigonométrica
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1 ------------------------------ / 16 16 \ |- ------- + -------|*csc(2*x) | 2 2 | \ sec (x) csc (x)/
$$\frac{1}{\left(- \frac{16}{\sec^{2}{\left(x \right)}} + \frac{16}{\csc^{2}{\left(x \right)}}\right) \csc{\left(2 x \right)}}$$
-1 ----------- 16*cot(2*x)
$$- \frac{1}{16 \cot{\left(2 x \right)}}$$
2*cot(x)
-----------------------------------------------------
/ 2 \
| / 2/x\\ 2/x\ |
| 16*|-1 + cot |-|| 64*cot |-| |
/ 2 \ | \ \2// \2/ |
\1 + cot (x)/*|- ------------------ + --------------|
| 2 2|
| / 2/x\\ / 2/x\\ |
| |1 + cot |-|| |1 + cot |-|| |
\ \ \2// \ \2// /
$$\frac{2 \cot{\left(x \right)}}{\left(- \frac{16 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{64 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}\right) \left(\cot^{2}{\left(x \right)} + 1\right)}$$
2 -sin (2*x) ----------- 8*sin(4*x)
$$- \frac{\sin^{2}{\left(2 x \right)}}{8 \sin{\left(4 x \right)}}$$
/pi \
-csc|-- - 2*x|
\2 /
---------------
16*csc(2*x)
$$- \frac{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}{16 \csc{\left(2 x \right)}}$$
2*tan(x)
----------------------------------------------------
/ 2 \
| / 2/x\\ 2/x\ |
| 16*|1 - tan |-|| 64*tan |-| |
/ 2 \ | \ \2// \2/ |
\1 + tan (x)/*|- ----------------- + --------------|
| 2 2|
| / 2/x\\ / 2/x\\ |
| |1 + tan |-|| |1 + tan |-|| |
\ \ \2// \ \2// /
$$\frac{2 \tan{\left(x \right)}}{\left(- \frac{16 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{64 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}$$
sin(2*x)
-------------------------
2 2
- 16*cos (x) + 16*sin (x)
$$\frac{\sin{\left(2 x \right)}}{16 \sin^{2}{\left(x \right)} - 16 \cos^{2}{\left(x \right)}}$$
/ pi\
-cos|2*x - --|
\ 2 /
---------------
16*cos(2*x)
$$- \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{16 \cos{\left(2 x \right)}}$$
1 ----------------------------------- / 16 16 \ |- ------------ + -------|*csc(2*x) | 2/pi \ 2 | | csc |-- - x| csc (x)| \ \2 / /
$$\frac{1}{\left(- \frac{16}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{16}{\csc^{2}{\left(x \right)}}\right) \csc{\left(2 x \right)}}$$
sin(2*x)
------------------------------
2/ pi\ 2
- 16*sin |x + --| + 16*sin (x)
\ 2 /
$$\frac{\sin{\left(2 x \right)}}{16 \sin^{2}{\left(x \right)} - 16 \sin^{2}{\left(x + \frac{\pi}{2} \right)}}$$
1 ---------------------------------------- / 16 16 \ / pi\ |- ------- + ------------|*sec|2*x - --| | 2 2/ pi\| \ 2 / | sec (x) sec |x - --|| \ \ 2 //
$$\frac{1}{\left(\frac{16}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} - \frac{16}{\sec^{2}{\left(x \right)}}\right) \sec{\left(2 x - \frac{\pi}{2} \right)}}$$
-sec(2*x)
----------------
/ pi\
16*sec|2*x - --|
\ 2 /
$$- \frac{\sec{\left(2 x \right)}}{16 \sec{\left(2 x - \frac{\pi}{2} \right)}}$$
-tan(2*x)
----------
16
$$- \frac{\tan{\left(2 x \right)}}{16}$$
/ pi\
cos|2*x - --|
\ 2 /
------------------------------
2 2/ pi\
- 16*cos (x) + 16*cos |x - --|
\ 2 /
$$\frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{- 16 \cos^{2}{\left(x \right)} + 16 \cos^{2}{\left(x - \frac{\pi}{2} \right)}}$$
cos(2*x - pi/2)/(-16*cos(x)^2 + 16*cos(x - pi/2)^2)