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