Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
x/(2-4*tan(x/2)^2+2*tan(x/2)^4)-2*tan(x/2)/(2-4*tan(x/2)^2+2*tan(x/2)^4)+2*tan(x/2)^3/(2-4*tan(x/2)^2+2*tan(x/2)^4)+x*tan(x/2)^4/(2-4*tan(x/2)^2+2*tan(x/2)^4)+2*x*tan(x/2)^2/(2-4*tan(x/2)^2+2*tan(x/2)^4)
¿Cómo vas a descomponer esta x/(2-4*tan(x/2)^2+2*tan(x/2)^4)-2*tan(x/2)/(2-4*tan(x/2)^2+2*tan(x/2)^4)+2*tan(x/2)^3/(2-4*tan(x/2)^2+2*tan(x/2)^4)+x*tan(x/2)^4/(2-4*tan(x/2)^2+2*tan(x/2)^4)+2*x*tan(x/2)^2/(2-4*tan(x/2)^2+2*tan(x/2)^4) expresión en fracciones?
Solución
Ha introducido
[src]
/x\ 3/x\ 4/x\ 2/x\
2*tan|-| 2*tan |-| x*tan |-| 2*x*tan |-|
x \2/ \2/ \2/ \2/
------------------------- - ------------------------- + ------------------------- + ------------------------- + -------------------------
2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\
2 - 4*tan |-| + 2*tan |-| 2 - 4*tan |-| + 2*tan |-| 2 - 4*tan |-| + 2*tan |-| 2 - 4*tan |-| + 2*tan |-| 2 - 4*tan |-| + 2*tan |-|
\2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/
$$\frac{2 x \tan^{2}{\left(\frac{x}{2} \right)}}{\left(2 - 4 \tan^{2}{\left(\frac{x}{2} \right)}\right) + 2 \tan^{4}{\left(\frac{x}{2} \right)}} + \left(\frac{x \tan^{4}{\left(\frac{x}{2} \right)}}{\left(2 - 4 \tan^{2}{\left(\frac{x}{2} \right)}\right) + 2 \tan^{4}{\left(\frac{x}{2} \right)}} + \left(\left(\frac{x}{\left(2 - 4 \tan^{2}{\left(\frac{x}{2} \right)}\right) + 2 \tan^{4}{\left(\frac{x}{2} \right)}} - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(2 - 4 \tan^{2}{\left(\frac{x}{2} \right)}\right) + 2 \tan^{4}{\left(\frac{x}{2} \right)}}\right) + \frac{2 \tan^{3}{\left(\frac{x}{2} \right)}}{\left(2 - 4 \tan^{2}{\left(\frac{x}{2} \right)}\right) + 2 \tan^{4}{\left(\frac{x}{2} \right)}}\right)\right)$$
x/(2 - 4*tan(x/2)^2 + 2*tan(x/2)^4) - 2*tan(x/2)/(2 - 4*tan(x/2)^2 + 2*tan(x/2)^4) + (2*tan(x/2)^3)/(2 - 4*tan(x/2)^2 + 2*tan(x/2)^4) + (x*tan(x/2)^4)/(2 - 4*tan(x/2)^2 + 2*tan(x/2)^4) + ((2*x)*tan(x/2)^2)/(2 - 4*tan(x/2)^2 + 2*tan(x/2)^4)
Simplificación general
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4/x\
x*tan |-|
3/x\ x /x\ 2/x\ \2/
tan |-| + - - tan|-| + x*tan |-| + ---------
\2/ 2 \2/ \2/ 2
--------------------------------------------
2 2
/ /x\\ / /x\\
|1 + tan|-|| *|-1 + tan|-||
\ \2// \ \2//
$$\frac{\frac{x \tan^{4}{\left(\frac{x}{2} \right)}}{2} + x \tan^{2}{\left(\frac{x}{2} \right)} + \frac{x}{2} + \tan^{3}{\left(\frac{x}{2} \right)} - \tan{\left(\frac{x}{2} \right)}}{\left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
(tan(x/2)^3 + x/2 - tan(x/2) + x*tan(x/2)^2 + x*tan(x/2)^4/2)/((1 + tan(x/2))^2*(-1 + tan(x/2))^2)
Respuesta numérica
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x/(2.0 + 2.0*tan(x/2)^4 - 4.0*tan(x/2)^2) + 2.0*tan(x/2)^3/(2.0 + 2.0*tan(x/2)^4 - 4.0*tan(x/2)^2) - 2.0*tan(x/2)/(2.0 + 2.0*tan(x/2)^4 - 4.0*tan(x/2)^2) + x*tan(x/2)^4/(2.0 + 2.0*tan(x/2)^4 - 4.0*tan(x/2)^2) + 2.0*x*tan(x/2)^2/(2.0 + 2.0*tan(x/2)^4 - 4.0*tan(x/2)^2)
x/(2.0 + 2.0*tan(x/2)^4 - 4.0*tan(x/2)^2) + 2.0*tan(x/2)^3/(2.0 + 2.0*tan(x/2)^4 - 4.0*tan(x/2)^2) - 2.0*tan(x/2)/(2.0 + 2.0*tan(x/2)^4 - 4.0*tan(x/2)^2) + x*tan(x/2)^4/(2.0 + 2.0*tan(x/2)^4 - 4.0*tan(x/2)^2) + 2.0*x*tan(x/2)^2/(2.0 + 2.0*tan(x/2)^4 - 4.0*tan(x/2)^2)
Denominador común
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3/x\ /x\ 2/x\
tan |-| - tan|-| + 2*x*tan |-|
x \2/ \2/ \2/
- + ------------------------------
2 4/x\ 2/x\
1 + tan |-| - 2*tan |-|
\2/ \2/
$$\frac{x}{2} + \frac{2 x \tan^{2}{\left(\frac{x}{2} \right)} + \tan^{3}{\left(\frac{x}{2} \right)} - \tan{\left(\frac{x}{2} \right)}}{\tan^{4}{\left(\frac{x}{2} \right)} - 2 \tan^{2}{\left(\frac{x}{2} \right)} + 1}$$
x/2 + (tan(x/2)^3 - tan(x/2) + 2*x*tan(x/2)^2)/(1 + tan(x/2)^4 - 2*tan(x/2)^2)
Potencias
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4 3 2
/ I*x -I*x \ / I*x -I*x \ / I*x -I*x \ / I*x -I*x \
| --- -----| | --- -----| | --- -----| | --- -----|
| 2 2 | | 2 2 | | 2 2 | | 2 2 |
x x*\- e + e / 2*I*\- e + e / 2*I*\- e + e / 2*x*\- e + e /
----------------------------------------------- + ------------------------------------------------------------------ - ------------------------------------------------------------------ - ----------------------------------------------------------------- - ------------------------------------------------------------------
4 2 / 4 2\ / 4 2\ / 4 2\ / 4 2\
/ I*x -I*x \ / I*x -I*x \ 4 | / I*x -I*x \ / I*x -I*x \ | 3 | / I*x -I*x \ / I*x -I*x \ | | / I*x -I*x \ / I*x -I*x \ | 2 | / I*x -I*x \ / I*x -I*x \ |
| --- -----| | --- -----| / I*x -I*x \ | | --- -----| | --- -----| | / I*x -I*x \ | | --- -----| | --- -----| | / I*x -I*x \ | | --- -----| | --- -----| | / I*x -I*x \ | | --- -----| | --- -----| |
| 2 2 | | 2 2 | | --- -----| | | 2 2 | | 2 2 | | | --- -----| | | 2 2 | | 2 2 | | | --- -----| | | 2 2 | | 2 2 | | | --- -----| | | 2 2 | | 2 2 | |
2*\- e + e / 4*\- e + e / | 2 2 | | 2*\- e + e / 4*\- e + e / | | 2 2 | | 2*\- e + e / 4*\- e + e / | | 2 2 | | 2*\- e + e / 4*\- e + e / | | 2 2 | | 2*\- e + e / 4*\- e + e / |
2 + -------------------- + -------------------- \e + e / *|2 + -------------------- + --------------------| \e + e / *|2 + -------------------- + --------------------| \e + e /*|2 + -------------------- + --------------------| \e + e / *|2 + -------------------- + --------------------|
4 2 | 4 2 | | 4 2 | | 4 2 | | 4 2 |
/ I*x -I*x \ / I*x -I*x \ | / I*x -I*x \ / I*x -I*x \ | | / I*x -I*x \ / I*x -I*x \ | | / I*x -I*x \ / I*x -I*x \ | | / I*x -I*x \ / I*x -I*x \ |
| --- -----| | --- -----| | | --- -----| | --- -----| | | | --- -----| | --- -----| | | | --- -----| | --- -----| | | | --- -----| | --- -----| |
| 2 2 | | 2 2 | | | 2 2 | | 2 2 | | | | 2 2 | | 2 2 | | | | 2 2 | | 2 2 | | | | 2 2 | | 2 2 | |
\e + e / \e + e / \ \e + e / \e + e / / \ \e + e / \e + e / / \ \e + e / \e + e / / \ \e + e / \e + e / /
$$\frac{x \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4} \left(\frac{2 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}} + \frac{4 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + 2\right)} - \frac{2 x \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2} \left(\frac{2 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}} + \frac{4 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + 2\right)} + \frac{x}{\frac{2 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}} + \frac{4 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + 2} - \frac{2 i \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{3}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{3} \left(\frac{2 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}} + \frac{4 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + 2\right)} - \frac{2 i \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right) \left(\frac{2 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}} + \frac{4 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + 2\right)}$$
/x\ 3/x\ 4/x\ 2/x\
2*tan|-| 2*tan |-| x*tan |-| 2*x*tan |-|
x \2/ \2/ \2/ \2/
------------------------- - ------------------------- + ------------------------- + ------------------------- + -------------------------
2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\
2 - 4*tan |-| + 2*tan |-| 2 - 4*tan |-| + 2*tan |-| 2 - 4*tan |-| + 2*tan |-| 2 - 4*tan |-| + 2*tan |-| 2 - 4*tan |-| + 2*tan |-|
\2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/
$$\frac{x \tan^{4}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{2 x \tan^{2}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{x}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{2 \tan^{3}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} - \frac{2 \tan{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2}$$
x/(2 - 4*tan(x/2)^2 + 2*tan(x/2)^4) - 2*tan(x/2)/(2 - 4*tan(x/2)^2 + 2*tan(x/2)^4) + 2*tan(x/2)^3/(2 - 4*tan(x/2)^2 + 2*tan(x/2)^4) + x*tan(x/2)^4/(2 - 4*tan(x/2)^2 + 2*tan(x/2)^4) + 2*x*tan(x/2)^2/(2 - 4*tan(x/2)^2 + 2*tan(x/2)^4)
Combinatoria
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/x\ 3/x\ 4/x\ 2/x\
x - 2*tan|-| + 2*tan |-| + x*tan |-| + 2*x*tan |-|
\2/ \2/ \2/ \2/
--------------------------------------------------
2 2
/ /x\\ / /x\\
2*|1 + tan|-|| *|-1 + tan|-||
\ \2// \ \2//
$$\frac{x \tan^{4}{\left(\frac{x}{2} \right)} + 2 x \tan^{2}{\left(\frac{x}{2} \right)} + x + 2 \tan^{3}{\left(\frac{x}{2} \right)} - 2 \tan{\left(\frac{x}{2} \right)}}{2 \left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
(x - 2*tan(x/2) + 2*tan(x/2)^3 + x*tan(x/2)^4 + 2*x*tan(x/2)^2)/(2*(1 + tan(x/2))^2*(-1 + tan(x/2))^2)
Unión de expresiones racionales
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/x\ 3/x\ 4/x\ 2/x\
x - 2*tan|-| + 2*tan |-| + x*tan |-| + 2*x*tan |-|
\2/ \2/ \2/ \2/
--------------------------------------------------
/ 4/x\ 2/x\\
2*|1 + tan |-| - 2*tan |-||
\ \2/ \2//
$$\frac{x \tan^{4}{\left(\frac{x}{2} \right)} + 2 x \tan^{2}{\left(\frac{x}{2} \right)} + x + 2 \tan^{3}{\left(\frac{x}{2} \right)} - 2 \tan{\left(\frac{x}{2} \right)}}{2 \left(\tan^{4}{\left(\frac{x}{2} \right)} - 2 \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
(x - 2*tan(x/2) + 2*tan(x/2)^3 + x*tan(x/2)^4 + 2*x*tan(x/2)^2)/(2*(1 + tan(x/2)^4 - 2*tan(x/2)^2))
Denominador racional
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/x\ 3/x\ 4/x\ 2/x\
x - 2*tan|-| + 2*tan |-| + x*tan |-| + 2*x*tan |-|
\2/ \2/ \2/ \2/
--------------------------------------------------
2/x\ 4/x\
2 - 4*tan |-| + 2*tan |-|
\2/ \2/
$$\frac{x \tan^{4}{\left(\frac{x}{2} \right)} + 2 x \tan^{2}{\left(\frac{x}{2} \right)} + x + 2 \tan^{3}{\left(\frac{x}{2} \right)} - 2 \tan{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2}$$
(x - 2*tan(x/2) + 2*tan(x/2)^3 + x*tan(x/2)^4 + 2*x*tan(x/2)^2)/(2 - 4*tan(x/2)^2 + 2*tan(x/2)^4)
Compilar la expresión
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/x\ 3/x\ 4/x\ 2/x\
2*tan|-| 2*tan |-| x*tan |-| 2*x*tan |-|
x \2/ \2/ \2/ \2/
------------------------- - ------------------------- + ------------------------- + ------------------------- + -------------------------
2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\
2 - 4*tan |-| + 2*tan |-| 2 - 4*tan |-| + 2*tan |-| 2 - 4*tan |-| + 2*tan |-| 2 - 4*tan |-| + 2*tan |-| 2 - 4*tan |-| + 2*tan |-|
\2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/
$$\frac{x \tan^{4}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{2 x \tan^{2}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{x}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{2 \tan^{3}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} - \frac{2 \tan{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2}$$
x/(2 - 4*tan(x/2)^2 + 2*tan(x/2)^4) - 2*tan(x/2)/(2 - 4*tan(x/2)^2 + 2*tan(x/2)^4) + 2*tan(x/2)^3/(2 - 4*tan(x/2)^2 + 2*tan(x/2)^4) + x*tan(x/2)^4/(2 - 4*tan(x/2)^2 + 2*tan(x/2)^4) + 2*x*tan(x/2)^2/(2 - 4*tan(x/2)^2 + 2*tan(x/2)^4)