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