Sr Examen
¿Cómo vas a descomponer esta cos(2*x)*x/(sin(x)-cos(x)) expresión en fracciones?
Solución
Ha introducido
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cos(2*x)*x --------------- sin(x) - cos(x)
$$\frac{x \cos{\left(2 x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}}$$
(cos(2*x)*x)/(sin(x) - cos(x))
Potencias
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/ -2*I*x 2*I*x\
|e e |
x*|------- + ------|
\ 2 2 /
-----------------------------------
I*x -I*x / -I*x I*x\
e e I*\- e + e /
- ---- - ----- - ------------------
2 2 2
$$\frac{x \left(\frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2}\right)}{- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} - \frac{e^{i x}}{2} - \frac{e^{- i x}}{2}}$$
x*(exp(-2*i*x)/2 + exp(2*i*x)/2)/(-exp(i*x)/2 - exp(-i*x)/2 - i*(-exp(-i*x) + exp(i*x))/2)
Combinatoria
[src]
-x*cos(2*x) ---------------- -sin(x) + cos(x)
$$- \frac{x \cos{\left(2 x \right)}}{- \sin{\left(x \right)} + \cos{\left(x \right)}}$$
-x*cos(2*x)/(-sin(x) + cos(x))
Parte trigonométrica
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___ -x*\/ 2 ----------- / pi\ sec|x - --| \ 4 /
$$- \frac{\sqrt{2} x}{\sec{\left(x - \frac{\pi}{4} \right)}}$$
___
x*\/ 2 *cos(2*x)
-----------------------------------------------------
___________ ___________
___ / ___ / ___
- \/ 2 *cos(x) + \/ 2 + \/ 2 *\/ 2 - \/ 2 *sin(x)
$$\frac{\sqrt{2} x \cos{\left(2 x \right)}}{\sqrt{2 - \sqrt{2}} \sqrt{\sqrt{2} + 2} \sin{\left(x \right)} - \sqrt{2} \cos{\left(x \right)}}$$
x ------------------------------- / 1 1 \ |----------- - ------|*sec(2*x) | / pi\ sec(x)| |sec|x - --| | \ \ 2 / /
$$\frac{x}{\left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(2 x \right)}}$$
x*cos(2*x)
---------------------
/ pi\
-cos(x) + cos|x - --|
\ 2 /
$$\frac{x \cos{\left(2 x \right)}}{- \cos{\left(x \right)} + \cos{\left(x - \frac{\pi}{2} \right)}}$$
___ 2 / 2/x pi\\ / 2 \
-x*\/ 2 *cos (x)*|1 + tan |- + --||*\1 - tan (x)/
\ \2 8 //
--------------------------------------------------
/ 2/x pi\\
2*|1 - tan |- + --||
\ \2 8 //
$$- \frac{\sqrt{2} x \left(1 - \tan^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(x \right)}}{2 \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right)}$$
___ / pi\
-x*\/ 2 *cos(2*x)*csc|-x + --|
\ 4 /
-------------------------------
2
$$- \frac{\sqrt{2} x \cos{\left(2 x \right)} \csc{\left(- x + \frac{\pi}{4} \right)}}{2}$$
___ -x*\/ 2 ----------- / pi\ csc|x + --| \ 4 /
$$- \frac{\sqrt{2} x}{\csc{\left(x + \frac{\pi}{4} \right)}}$$
___ /pi \
-x*\/ 2 *sin|-- + 2*x|
\2 /
-----------------------
/ 3*pi\
2*sin|x + ----|
\ 4 /
$$- \frac{\sqrt{2} x \sin{\left(2 x + \frac{\pi}{2} \right)}}{2 \sin{\left(x + \frac{3 \pi}{4} \right)}}$$
___ / 2/x pi\\ / 2 \
-x*\/ 2 *|1 + cot |- + --||*\-1 + cot (x)/
\ \2 8 //
-------------------------------------------
/ 2 \ / 2/x pi\\
2*\1 + cot (x)/*|-1 + cot |- + --||
\ \2 8 //
$$- \frac{\sqrt{2} x \left(\cot^{2}{\left(x \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right)}{2 \left(\cot^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right)}$$
___ /x pi\
-2*x*\/ 2 *tan|- + --|
\2 8 /
----------------------
2/x pi\
1 + tan |- + --|
\2 8 /
$$- \frac{2 \sqrt{2} x \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}$$
x -------------------------- / 1 1 \ |------ - ------|*sec(2*x) \csc(x) sec(x)/
$$\frac{x}{\left(- \frac{1}{\sec{\left(x \right)}} + \frac{1}{\csc{\left(x \right)}}\right) \sec{\left(2 x \right)}}$$
___ 2/x pi\ /x pi\
-2*x*\/ 2 *sin |- + --|*cot|- + --|
\2 8 / \2 8 /
$$- 2 \sqrt{2} x \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)}$$
___ / pi\
-x*\/ 2 *csc|-x + --|
\ 4 /
----------------------
/pi \
2*csc|-- - 2*x|
\2 /
$$- \frac{\sqrt{2} x \csc{\left(- x + \frac{\pi}{4} \right)}}{2 \csc{\left(- 2 x + \frac{\pi}{2} \right)}}$$
___
-x*\/ 2 *cos(2*x)
------------------
/ pi\
2*cos|x + --|
\ 4 /
$$- \frac{\sqrt{2} x \cos{\left(2 x \right)}}{2 \cos{\left(x + \frac{\pi}{4} \right)}}$$
/pi \
x*sin|-- + 2*x|
\2 /
----------------------
/ pi\
- sin|x + --| + sin(x)
\ 2 /
$$\frac{x \sin{\left(2 x + \frac{\pi}{2} \right)}}{\sin{\left(x \right)} - \sin{\left(x + \frac{\pi}{2} \right)}}$$
/ 2 \
x*\-1 + cot (x)/
--------------------------------------------
/ 2/x\ /x\ \
| -1 + cot |-| 2*cot|-| |
/ 2 \ | \2/ \2/ |
\1 + cot (x)/*|- ------------ + -----------|
| 2/x\ 2/x\|
| 1 + cot |-| 1 + cot |-||
\ \2/ \2//
$$\frac{x \left(\cot^{2}{\left(x \right)} - 1\right)}{\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(x \right)} + 1\right)}$$
___ / pi\
-x*\/ 2 *sin|x + --|
\ 4 /
$$- \sqrt{2} x \sin{\left(x + \frac{\pi}{4} \right)}$$
/ 2 \
x*\1 - tan (x)/
-------------------------------------------
/ 2/x\ /x\ \
| 1 - tan |-| 2*tan|-| |
/ 2 \ | \2/ \2/ |
\1 + tan (x)/*|- ----------- + -----------|
| 2/x\ 2/x\|
| 1 + tan |-| 1 + tan |-||
\ \2/ \2//
$$\frac{x \left(1 - \tan^{2}{\left(x \right)}\right)}{\left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}$$
___ /x pi\
-2*x*\/ 2 *cot|- + --|
\2 8 /
----------------------
2/x pi\
1 + cot |- + --|
\2 8 /
$$- \frac{2 \sqrt{2} x \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}$$
x ------------------------------------ / 1 1 \ /pi \ |------ - -----------|*csc|-- - 2*x| |csc(x) /pi \| \2 / | csc|-- - x|| \ \2 //
$$\frac{x}{\left(- \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(- 2 x + \frac{\pi}{2} \right)}}$$
___ / 2/x pi\\ / 2 \
-x*\/ 2 *|1 + tan |- + --||*\1 - tan (x)/
\ \2 8 //
------------------------------------------
/ 2 \ / 2/x pi\\
2*\1 + tan (x)/*|1 - tan |- + --||
\ \2 8 //
$$- \frac{\sqrt{2} x \left(1 - \tan^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right)}{2 \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}$$
___ / pi\
-x*\/ 2 *cos(2*x)*sec|x + --|
\ 4 /
------------------------------
2
$$- \frac{\sqrt{2} x \cos{\left(2 x \right)} \sec{\left(x + \frac{\pi}{4} \right)}}{2}$$
___ / pi\
-x*\/ 2 *cos|x - --|
\ 4 /
$$- \sqrt{2} x \cos{\left(x - \frac{\pi}{4} \right)}$$
___ / pi\
-x*\/ 2 *sec|x + --|
\ 4 /
---------------------
2*sec(2*x)
$$- \frac{\sqrt{2} x \sec{\left(x + \frac{\pi}{4} \right)}}{2 \sec{\left(2 x \right)}}$$
-x*sqrt(2)*sec(x + pi/4)/(2*sec(2*x))
Denominador común
[src]
-x*cos(2*x) ---------------- -sin(x) + cos(x)
$$- \frac{x \cos{\left(2 x \right)}}{- \sin{\left(x \right)} + \cos{\left(x \right)}}$$
-x*cos(2*x)/(-sin(x) + cos(x))